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12.2: Powers and Roots

12.2: Powers and Roots


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Table B1

(n)(n^{2})(sqrt{n})(n^{3})(sqrt[3]{n})
11111
241.41421481.259921
391.732051271.442250
4162641.587401
5252.2360681251.709976
6362.4494902161.817121
7492.6457513431.912931
8642.8284275122
98137292.080084
101003.1622781,0002.154435
1112133166251,3311.223980
121443.4641021,7282.289428
131693.6055512,1972.351335
141963.7416572,7442.410142
152253.8729833,3752,466212
1625644,0962.519842
172894.1231064,9132.571282
183244.2426415,8322.620741
193614.3588996,8592.668402
204004.4721368,0002.714418
214414.5825769,2612.758924
224844.69041610,6482.802039
235294.79583212,1672.843867
245764.89897913,8242.884499
25625515,6252.924018
266765.09902017,5762.962496
277295.19615219,6833
287845.29150321,9523.036589
298415.38516524,3893.072317
309005.47722627,0003.107233
319615.56776429,7913.141381
321,0245.65685432,7683.17482
331,0895.74456335,9373.207534
341,1565.83095239,3043.239612
351,2255.91608042,8753.271066
361,296646,6563.301927
371,3696.08276350,6533.332222
381,444616441454,8723.361975
391,5216.24499859,3193.391211
401,6006.32455564,0003.419952
411,6816.40312468,9213.448217
421.7646.48074174,0883.476027
431.8496.55743979,5073.503398
441,9366.63325085,1843.530348
452,0256.70820491,1253.556893
462,1166.78233097,3363.583048
472,2096.855655103,8233.608826
482,3046.928203110,5923.6324241
492,4017117,6493.659306
502,5007.071068125,0003.684031
512,6017.141428132,6513.708430
522,7047.211103140,6083.732511
532,8097.280110148,8773.756286
542,9167.348469157,4643.779763
553,0257.416198166,3753.802952
563,1367.483315175,6163.825862
573,2497.549834185,1933.848501
583,3647.615773195,1123.870877
593,4817.681146205,3793.892996
603,6007.745967216,0003.914868
613,7217.810250226,9813.936497
623,8447.874008238,3283.957892
633,9697.937254250,0473.979057
644,0968262,1444
654,2258.062258274,6254.020726
666,3568.124038287,4964.041240
674,4898.185353300,7634.061548
684,6248.246211314,4324.081655
694,7618.306624328,5094.101566
704,9008.366600343,0004.121285
715,0418.426150357,9114.140818
725,1848.485281389,0174.179339
735,3298.544004389,0174.179339
745,4768.602325405,2244.198336
755,6258.660254421,8754.217163
765,7768.17798438,9764.235824
775.9298774964456,5334.254321
786,0848.831761474,5524.272659
796,2418.888194493,0394.290840
806,4008.944272512,0004.308869
816,5619531,4414.326749
826,7249.055385551,3684.344481
836,8899.110434571,7874.362071
847,0569.165151592,7044.379519
857,2259.219544614,1254.396830
867,3969.273618636,0564.414005
877,5699.327379658,5034.431048
887,7449.380832681,4724.447960
897,8218.433981704,9694.464745
908,1009.486833729,0004.481405
918,2819.539392753,5714.497941
928,4649.591663778,6884.514357
938,6499.643651804,3574.530655
948,8369.695360830,5844.546836
959,0259.746794857,3754.562903
969,2169.797959884,7364.578857
979,4099.848858912,6734.594701
989,6049.899495941,1924.610436
999,8019.949874970,2994.62065
10010,000101,000,0004.641589

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Contents

The squares (sequence A000290 in the OEIS) smaller than 60 2 = 3600 are:

The difference between any perfect square and its predecessor is given by the identity n 2 − (n − 1) 2 = 2n − 1 . Equivalently, it is possible to count square numbers by adding together the last square, the last square's root, and the current root, that is, n 2 = (n − 1) 2 + (n − 1) + n .

The number m is a square number if and only if one can arrange m points in a square:

m = 1 2 = 1
m = 2 2 = 4
m = 3 2 = 9
m = 4 2 = 16
m = 5 2 = 25

The expression for the n th square number is n 2 . This is also equal to the sum of the first n odd numbers as can be seen in the above pictures, where a square results from the previous one by adding an odd number of points (shown in magenta). The formula follows:

For example, 5 2 = 25 = 1 + 3 + 5 + 7 + 9 .

There are several recursive methods for computing square numbers. For example, the n th square number can be computed from the previous square by n 2 = (n − 1) 2 + (n − 1) + n = (n − 1) 2 + (2n − 1) . Alternatively, the n th square number can be calculated from the previous two by doubling the (n − 1) th square, subtracting the (n − 2) th square number, and adding 2, because n 2 = 2(n − 1) 2 − (n − 2) 2 + 2 . For example,

2 × 5 2 − 4 2 + 2 = 2 × 25 − 16 + 2 = 50 − 16 + 2 = 36 = 6 2 .

A square number is also the sum of two consecutive triangular numbers. The sum of two consecutive square numbers is a centered square number. Every odd square is also a centered octagonal number.

Another property of a square number is that (except 0) it has an odd number of positive divisors, while other natural numbers have an even number of positive divisors. An integer root is the only divisor that pairs up with itself to yield the square number, while other divisors come in pairs.

Lagrange's four-square theorem states that any positive integer can be written as the sum of four or fewer perfect squares. Three squares are not sufficient for numbers of the form 4 k (8m + 7) . A positive integer can be represented as a sum of two squares precisely if its prime factorization contains no odd powers of primes of the form 4k + 3 . This is generalized by Waring's problem.

In base 10, a square number can end only with digits 0, 1, 4, 5, 6 or 9, as follows:

  • if the last digit of a number is 0, its square ends in 0 (in fact, the last two digits must be 00)
  • if the last digit of a number is 1 or 9, its square ends in 1
  • if the last digit of a number is 2 or 8, its square ends in 4
  • if the last digit of a number is 3 or 7, its square ends in 9
  • if the last digit of a number is 4 or 6, its square ends in 6 and
  • if the last digit of a number is 5, its square ends in 5 (in fact, the last two digits must be 25).

In base 12, a square number can end only with square digits (like in base 12, a prime number can end only with prime digits or 1), that is, 0, 1, 4 or 9, as follows:

  • if a number is divisible both by 2 and by 3 (that is, divisible by 6), its square ends in 0
  • if a number is divisible neither by 2 nor by 3, its square ends in 1
  • if a number is divisible by 2, but not by 3, its square ends in 4 and
  • if a number is not divisible by 2, but by 3, its square ends in 9.

Similar rules can be given for other bases, or for earlier digits (the tens instead of the units digit, for example). [ citation needed ] All such rules can be proved by checking a fixed number of cases and using modular arithmetic.

In general, if a prime p divides a square number m then the square of p must also divide m if p fails to divide m / p , then m is definitely not square. Repeating the divisions of the previous sentence, one concludes that every prime must divide a given perfect square an even number of times (including possibly 0 times). Thus, the number m is a square number if and only if, in its canonical representation, all exponents are even.

Squarity testing can be used as alternative way in factorization of large numbers. Instead of testing for divisibility, test for squarity: for given m and some number k , if k 2 − m is the square of an integer n then kn divides m . (This is an application of the factorization of a difference of two squares.) For example, 100 2 − 9991 is the square of 3, so consequently 100 − 3 divides 9991. This test is deterministic for odd divisors in the range from kn to k + n where k covers some range of natural numbers k ≥ m . >.>

A square number cannot be a perfect number.

The sum of the n first square numbers is

The first values of these sums, the square pyramidal numbers, are: (sequence A000330 in the OEIS)

0, 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, 1015, 1240, 1496, 1785, 2109, 2470, 2870, 3311, 3795, 4324, 4900, 5525, 6201.


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Download the latest TWRP image file (.img) from the download link and boot TWRP. Go to install and find and select the Images. button. Browse to the image that you downloaded and select it. Choose recovery and swipe to flash.

You will need to download and install Odin on your PC. Once you have Odin installed, download the proper .tar file for your device from the download link above. Power off your device. Turn on your device using the proper key combo to get into download mode and attach the device to your computer via a USB cable. Use the PDA tab or button to browse to and select the tar file that you downloaded and flash the device.

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12.2: Powers and Roots

SQUARE ROOT CALCULATOR

Square Roots Calculator and Square Calculator

Looking for a faster, more convenient way to find square and square roots when doing homework or for mathematical applications? Check out our square roots calculator and square calculator.

What is a Square Calculator?

Our square calculator is aptly named: it is simply an online resource used to calculate the square of a number. A square is defined as the product of any number multiplied by itself (x2). For example, the square of 12 is 144 (the product of 12 times 12).

What is a Square Roots Calculator?

A square roots calculator finds the number that, when multiplied by itself, would give you the number you are starting out with. For example, the square root of 144 is 12, because 12 times 12 equals 144. Of course, -12 times -12 is also 144. Therefore, every number actually has two square roots. One thing to be aware of when using any calculator with square root functionality is that it will probably only give you the primary square root—that is, the positive square root. For most applications, the positive root is all you need. However, for mathematical problems remember that the second negative square root is also needed for the correct answer!

Online vs Handheld Calculators

While many of us grew up using a handheld calculator with square root keys, now we also have an online calculator square root option. In most cases, the online calculator is more useful. You might not always have a handheld calculator at the ready when you need to know a square or square root, but you probably do have your smartphone or tablet, which will allow you to access the online tool anywhere you have wifi or data service. Plus, most online calculator square root tools are easier to use than their handheld counterparts. Instead of worrying about pushing the calculator keys in the proper sequence, all you have to do online is type in your number and click a single button.

Square Root Calculator History
Modern technology has made many aspects of our lives much easier, including mathematics. Many mathematical tasks that used to take hours of tedious calculations can now be accomplished in just a few minutes using a computer or calculator. In order to better appreciate what calculators save us from, it is interesting to look at the history of modern tools like the square root calculator.

Simplest Manual Square Root Method
The simplest method for finding square roots manually is just to guess. Pick a number, square it, and see if the result is too low or too high. Of course, it will be easier if you make an educated guess based on the closest perfect squares (numbers that can be expressed as a whole integer times itself). For example, if you want to find the square root of 18, begin by finding the closest perfect squares, which are 16 and 25. The square roots of these numbers are 4 and 5. So, because 18 is between 16 and 25, we know its square root must be between their square roots, that is, between 4 and 5. It should be closer to 4 because 18 is closer to 16. Take a guess…let&rsquos say 4.3. 4.3 times 4.3 is 18.49. That&rsquos more than 18. So make your next guess a bit lower. 4.2 squared is 17.64…too low! Now try something halfway between the two guesses: 4.25. 4.25 squared is 18.0625. Close enough!

Oldest Manual Square Root Method

The oldest method for finding the square root of a number is known as the &ldquoBabylonian Method&rdquo after the civilization that historians believe first used this method. It is also sometimes called &ldquoHeron&rsquos Method&rdquo after the Greek mathematician that recorded the first detailed description of the method. This method is basically a fancier way of doing the simple guessing method. It uses an algorithm to calculate the square root based on the average of an underestimate and overestimate of the root.

Modern Square Root Calculator

Of course, when you want a fast and accurate square root calculated to multiple decimal places, a modern square root calculator is by far your best option. These types of calculators first began to crop up in the 1970s, and a calculator with all the bells and whistles could be quite expensive. Nowadays, however, you can find square roots even more easily without special equipment, just by using an online calculator.


You must remember that $x = -sqrt$ in any problem where $x o, -infty$, since you’re then automatically looking at negative values of x.

The problems below illustrate, starting with part (b) of the first one.

Note that in the last step, we used the fact that $displaystylefrac<2> = 0>$.



We can verify the result with a quick look at the graph of the function. Note that the horizontal line $y = sqrt<5>$ is a horizontal asymptote for this graph.

To obtain that result, we again use our usual “trick” of dividing the numerator and the denominator by the largest term in the denominator, which here is $x$.

Notice that we obtained a negative number as our answer, which matches our quick initial reasoning above.

We can verify the result with a quick look at the graph of the function. Note that the horizontal line $y = -sqrt<5>$ is a horizontal asymptote for this graph.

To proceed, we’ll use the same approach we used earlier when evaluating limits that had square roots in them: we’ll rationalize the expression by multiplying by its conjugate $sqrt + x$ divided by itself:

[ egin
lim_left(sqrt – x ight) &= lim_left(frac – x> <1>cdot frac + x> + x> ight) [8px] &= lim_frac ight)^2 + xsqrt -x sqrt -x^2 > + x> [8px] &= lim_frac<(x^2 + x) -x^2> + x> [8px] &= lim_frac + x>
end ] Let’s now use our usual trick of dividing the numerator and the denominator by the largest power in the denominator. That power is $x$: while there is an $x^2$ present, it is under a square root $left(sqrt ight)$, and so its effective power is $x^1$.

This limit is unexpected, at least to us! But you can check a few numbers to see how it works:
[ egin
f(x) &= sqrt – x [8px] f(10) &= sqrt <100 + 10>– 10 approx 10.488 – 10 = 0.488 [8px] f(20) &= sqrt <400 + 20>– 20 approx 20.494 – 20 = 0.494 [8px] f(100) &= sqrt <10,000 + 100>– 100 approx 100.499 -100 = 0.499
end ]


To proceed, we’ll use the same approach we used earlier when evaluating limits that had square roots in them: we’ll rationalize the expression by multiplying by its conjugate $sqrt> + sqrt$ divided by itself:

Note that toward the end, we used the fact that $displaystylefrac<1>> = 0>$.

This limit is unexpected, at least to us! But you can check a few numbers to see how it works:
[ egin
f(x) &= sqrt> – sqrt [8px] f(100) &= sqrt<100 + sqrt<100>> – sqrt <100>approx 10.48 – 10 = 0.48 [8px] f(10,000) &= sqrt <10,000 + 100>– 100 approx 100.499 – 100 = 0.499 [8px] end ]

To proceed, we’ll use the same approach we used earlier when evaluating limits that had square roots in them: we’ll rationalize the expression by multiplying by its conjugate $sqrt + ax$ divided by itself:

Let’s now use our usual trick of dividing the numerator and the denominator by the largest power in the denominator. That power is $x:$ while there is an $x^2$ present, it is under a square root $left(sqrt ight)$, and so its effective power is $x^1.$

Since we’re looking at $x o infty$ we’re interested only in positive values of $x$, and so we have $x = sqrt$.

Let’s now use our usual trick of dividing the numerator and the denominator by the largest power in the denominator. That power is $x$: while there is an $x^2$ present, it is under a square root $left(sqrt ight)$, and so its effective power is $x^1$.

Since we’re looking at $x o infty$ we’re interested only in positive values of $x$, and so we have $x = sqrt.$

To proceed, we’ll use the same approach we used earlier when evaluating limits that had square roots in them: we’ll rationalize the expression by multiplying by its conjugate $x – sqrt$ divided by itself:

Let’s now use our usual trick of dividing the numerator and the denominator by the largest power in the denominator. That power is $x:$ while there is an $x^2$ present, it is under a square root $left(sqrt ight)$, and so its effective power is $x^1$.

Note that in the second to last line, we used the fact that $displaystyle frac = 0 >.$

As is so often the case, factoring provides a way forward: notice that we can pull a $sqrt$ out of both terms:

Step 1: As we have in the problems above, we multiply the expression by its conjugate divided by itself:
egin
lim_left( sqrt>> – sqrt ight) &= lim_left(sqrt>> – sqrt ight)cdot frac>> + sqrt>>> + sqrt> [8px] &= lim_ frac>> ight)left(sqrt>> ight) + cancel>> – sqrt ight)left(sqrt ight)> – cancel ight)left(sqrt>> – sqrt ight)> + left( -sqrt ight)left(sqrt ight) >>> + sqrt>[8px] &=lim_ frac> ight) -x>>> + sqrt> [8px] &= lim_frac>>>> + sqrt>
end
With this rewritten expression, you might be able to look at it and see that the numerator is dominated by the (first) $sqrt$ term, while the numerator is dominated equally by two factors of $sqrt$, and so the limit will be $dfrac<1><2>.$


Want access to all of our Calculus problems and solutions? Buy full access now — it’s quick and easy!

limit of square root of x^2+1 – square root of x+1 as x approaches infinity

We’ve updated the page to provide the solution to this question as Practice Problem #8 above. It’s a good problem, and we hope we’ve been able to help you (and future students) solve it successfully for yourself.

We’ve updated the page to provide the solution to this question as Practice Problem #8 above. It’s a good problem, and we hope we’ve been able to help you (and future students) solve it successfully for yourself.

root of x ( root of x+c – root x ) at lim x tends to infinite

Thanks for asking! (Note that you may need to refresh your browser page for the math below to display properly.)

It turns out there are two strategic moves you have to make to find this limit.

First, as discussed above, any time we see a two square roots subtracted from each other, we automatically multiply by the conjugate
[1= frac + sqrt> + sqrt>]
So
egin
lim_sqrt left( sqrt – sqrt ight) &= lim_ sqrtleft[left( sqrt – sqrt ight) cdot left( frac + sqrt> + sqrt> ight) ight] [8px]
&= lim_sqrt left[ frac<(x+c) +sqrtsqrt – sqrtsqrt-x > + sqrt> ight] [8px]
&= lim_sqrt left[ frac + sqrt> ight] [8px]
end
The other strategic move we need to make is a “trick” we developed earlier : find the largest power of $x$ in the denominator, and then factor it out. Here, that power is $sqrt$:
egin
phantom< lim_sqrt left( sqrt – sqrt ight)> &= lim_sqrt left[ fracleft( sqrt<1+frac>> + 1 ight)> ight] [8px]
&= lim_frac>> left[ frac>> + 1> ight] [8px]
&= lim_ frac< sqrt<1+frac>> + 1>
end
Now, since $lim_ dfrac> = 0,$ when we take the limit we have
egin
phantom< lim_sqrt left( sqrt – sqrt ight)> &= frac< sqrt<1+cancelto<0>< frac>>> + 1> [8px]
&= frac <2>quad cmark
end
Whew! : )


Leaves

Leaves are the main sites for photosynthesis: the process by which plants synthesize food. Most leaves are usually green, due to the presence of chlorophyll in the leaf cells. However, some leaves may have different colors, caused by other plant pigments that mask the green chlorophyll.

The thickness, shape, and size of leaves are adapted to the environment. Each variation helps a plant species maximize its chances of survival in a particular habitat. Usually, the leaves of plants growing in tropical rainforests have larger surface areas than those of plants growing in deserts or very cold conditions, which are likely to have a smaller surface area to minimize water loss.

Structure of a Typical Leaf

Figure 13. Deceptively simple in appearance, a leaf is a highly efficient structure.

Each leaf typically has a leaf blade called the lamina, which is also the widest part of the leaf. Some leaves are attached to the plant stem by a petiole. Leaves that do not have a petiole and are directly attached to the plant stem are called sessile leaves. Small green appendages usually found at the base of the petiole are known as stipules. Most leaves have a midrib, which travels the length of the leaf and branches to each side to produce veins of vascular tissue. The edge of the leaf is called the margin. Figure 13 shows the structure of a typical eudicot leaf.

Within each leaf, the vascular tissue forms veins. The arrangement of veins in a leaf is called the venation pattern. Monocots and dicots differ in their patterns of venation (Figure 14). Monocots have parallel venation the veins run in straight lines across the length of the leaf without converging at a point. In dicots, however, the veins of the leaf have a net-like appearance, forming a pattern known as reticulate venation. One extant plant, the Ginkgo biloba, has dichotomous venation where the veins fork.

Figure 14. (a) Tulip (Tulipa), a monocot, has leaves with parallel venation. The netlike venation in this (b) linden (Tilia cordata) leaf distinguishes it as a dicot. The (c) Ginkgo biloba tree has dichotomous venation. (credit a photo: modification of work by “Drewboy64”/Wikimedia Commons credit b photo: modification of work by Roger Griffith credit c photo: modification of work by “geishaboy500″/Flickr credit abc illustrations: modification of work by Agnieszka Kwiecień)

Leaf Arrangement

The arrangement of leaves on a stem is known as phyllotaxy. The number and placement of a plant’s leaves will vary depending on the species, with each species exhibiting a characteristic leaf arrangement. Leaves are classified as either alternate, spiral, or opposite. Plants that have only one leaf per node have leaves that are said to be either alternate—meaning the leaves alternate on each side of the stem in a flat plane—or spiral, meaning the leaves are arrayed in a spiral along the stem. In an opposite leaf arrangement, two leaves arise at the same point, with the leaves connecting opposite each other along the branch. If there are three or more leaves connected at a node, the leaf arrangement is classified as whorled.

Leaf Form

Leaves may be simple or compound (Figure 15). In a simple leaf, the blade is either completely undivided—as in the banana leaf—or it has lobes, but the separation does not reach the midrib, as in the maple leaf. In a compound leaf, the leaf blade is completely divided, forming leaflets, as in the locust tree. Each leaflet may have its own stalk, but is attached to the rachis. A palmately compound leaf resembles the palm of a hand, with leaflets radiating outwards from one point Examples include the leaves of poison ivy, the buckeye tree, or the familiar houseplant Schefflera sp. (common name “umbrella plant”). Pinnately compound leaves take their name from their feather-like appearance the leaflets are arranged along the midrib, as in rose leaves (Rosa sp.), or the leaves of hickory, pecan, ash, or walnut trees.

Figure 15. Leaves may be simple or compound. In simple leaves, the lamina is continuous. The (a) banana plant (Musa sp.) has simple leaves. In compound leaves, the lamina is separated into leaflets. Compound leaves may be palmate or pinnate. In (b) palmately compound leaves, such as those of the horse chestnut (Aesculus hippocastanum), the leaflets branch from the petiole. In (c) pinnately compound leaves, the leaflets branch from the midrib, as on a scrub hickory (Carya floridana). The (d) honey locust has double compound leaves, in which leaflets branch from the veins. (credit a: modification of work by “BazzaDaRambler”/Flickr credit b: modification of work by Roberto Verzo credit c: modification of work by Eric Dion credit d: modification of work by Valerie Lykes)

Leaf Structure and Function

The outermost layer of the leaf is the epidermis it is present on both sides of the leaf and is called the upper and lower epidermis, respectively. Botanists call the upper side the adaxial surface (or adaxis) and the lower side the abaxial surface (or abaxis). The epidermis helps in the regulation of gas exchange. It contains stomata (Figure 16): openings through which the exchange of gases takes place. Two guard cells surround each stoma, regulating its opening and closing.

Figure 16. Visualized at 500x with a scanning electron microscope, several stomata are clearly visible on (a) the surface of this sumac (Rhus glabra) leaf. At 5,000x magnification, the guard cells of (b) a single stoma from lyre-leaved sand cress (Arabidopsis lyrata) have the appearance of lips that surround the opening. In this (c) light micrograph cross-section of an A. lyrata leaf, the guard cell pair is visible along with the large, sub-stomatal air space in the leaf. (credit: modification of work by Robert R. Wise part c scale-bar data from Matt Russell)

The epidermis is usually one cell layer thick however, in plants that grow in very hot or very cold conditions, the epidermis may be several layers thick to protect against excessive water loss from transpiration. A waxy layer known as the cuticle covers the leaves of all plant species. The cuticle reduces the rate of water loss from the leaf surface. Other leaves may have small hairs (trichomes) on the leaf surface. Trichomes help to deter herbivory by restricting insect movements, or by storing toxic or bad-tasting compounds they can also reduce the rate of transpiration by blocking air flow across the leaf surface (Figure 17).

Figure 17. Trichomes give leaves a fuzzy appearance as in this (a) sundew (Drosera sp.). Leaf trichomes include (b) branched trichomes on the leaf of Arabidopsis lyrata and (c) multibranched trichomes on a mature Quercus marilandica leaf. (credit a: John Freeland credit b, c: modification of work by Robert R. Wise scale-bar data from Matt Russell)

Below the epidermis of dicot leaves are layers of cells known as the mesophyll, or “middle leaf.” The mesophyll of most leaves typically contains two arrangements of parenchyma cells: the palisade parenchyma and spongy parenchyma (Figure 18). The palisade parenchyma (also called the palisade mesophyll) has column-shaped, tightly packed cells, and may be present in one, two, or three layers. Below the palisade parenchyma are loosely arranged cells of an irregular shape. These are the cells of the spongy parenchyma (or spongy mesophyll). The air space found between the spongy parenchyma cells allows gaseous exchange between the leaf and the outside atmosphere through the stomata. In aquatic plants, the intercellular spaces in the spongy parenchyma help the leaf float. Both layers of the mesophyll contain many chloroplasts. Guard cells are the only epidermal cells to contain chloroplasts.

In the leaf drawing (Figure 18a), the central mesophyll is sandwiched between an upper and lower epidermis. The mesophyll has two layers: an upper palisade layer comprised of tightly packed, columnar cells, and a lower spongy layer, comprised of loosely packed, irregularly shaped cells. Stomata on the leaf underside allow gas exchange. A waxy cuticle covers all aerial surfaces of land plants to minimize water loss. These leaf layers are clearly visible in the scanning electron micrograph (Figure 18b). The numerous small bumps in the palisade parenchyma cells are chloroplasts. Chloroplasts are also present in the spongy parenchyma, but are not as obvious. The bumps protruding from the lower surface of the leave are glandular trichomes, which differ in structure from the stalked trichomes in Figure 17.

Figure 18. (a) Leaf drawing (b) Scanning electron micrograph of a leaf. (credit b: modification of work by Robert R. Wise)

Figure 19. This scanning electron micrograph shows xylem and phloem in the leaf vascular bundle from the lyre-leaved sand cress (Arabidopsis lyrata). (credit: modification of work by Robert R. Wise scale-bar data from Matt Russell)

Like the stem, the leaf contains vascular bundles composed of xylem and phloem (Figure 19). The xylem consists of tracheids and vessels, which transport water and minerals to the leaves. The phloem transports the photosynthetic products from the leaf to the other parts of the plant. A single vascular bundle, no matter how large or small, always contains both xylem and phloem tissues.

Leaf Adaptations

Coniferous plant species that thrive in cold environments, like spruce, fir, and pine, have leaves that are reduced in size and needle-like in appearance. These needle-like leaves have sunken stomata and a smaller surface area: two attributes that aid in reducing water loss. In hot climates, plants such as cacti have leaves that are reduced to spines, which in combination with their succulent stems, help to conserve water. Many aquatic plants have leaves with wide lamina that can float on the surface of the water, and a thick waxy cuticle on the leaf surface that repels water.

Watch “The Pale Pitcher Plant” episode of the video series Plants Are Cool, Too, a Botanical Society of America video about a carnivorous plant species found in Louisiana.


In Summary: Leaves

Leaves are the main site of photosynthesis. A typical leaf consists of a lamina (the broad part of the leaf, also called the blade) and a petiole (the stalk that attaches the leaf to a stem). The arrangement of leaves on a stem, known as phyllotaxy, enables maximum exposure to sunlight. Each plant species has a characteristic leaf arrangement and form. The pattern of leaf arrangement may be alternate, opposite, or spiral, while leaf form may be simple or compound. Leaf tissue consists of the epidermis, which forms the outermost cell layer, and mesophyll and vascular tissue, which make up the inner portion of the leaf. In some plant species, leaf form is modified to form structures such as tendrils, spines, bud scales, and needles.


12.2: Powers and Roots

Roots of Quadratic Equations and the Quadratic Formula

In this section, we will learn how to find the root(s) of a quadratic equation. Roots are also called x-intercepts or zeros. A quadratic function is graphically represented by a parabola with vertex located at the origin, below the x-axis, or above the x-axis. Therefore, a quadratic function may have one, two, or zero roots.

When we are asked to solve a quadratic equation, we are really being asked to find the roots. We have already seen that completing the square is a useful method to solve quadratic equations. This method can be used to derive the quadratic formula, which is used to solve quadratic equations. In fact, the roots of the function,

are given by the quadratic formula. The roots of a function are the x-intercepts. By definition, the y-coordinate of points lying on the x-axis is zero. Therefore, to find the roots of a quadratic function, we set f (x) = 0, and solve the equation,

We can do this by completing the square as,

Solving for x and simplifying we have,

Thus, the roots of a quadratic function are given by,

This formula is called the quadratic formula, and its derivation is included so that you can see where it comes from. We call the term b 2 &minus4ac the discriminant. The discriminant is important because it tells you how many roots a quadratic function has. Specifically, if

1. b 2 &minus4ac < 0 There are no real roots.

2. b 2 &minus4ac = 0 There is one real root.

3. b 2 &minus4ac > 0 There are two real roots.

We will examine each case individually.

Case 1: No Real Roots

If the discriminant of a quadratic function is less than zero, that function has no real roots, and the parabola it represents does not intersect the x-axis. Since the quadratic formula requires taking the square root of the discriminant, a negative discriminant creates a problem because the square root of a negative number is not defined over the real line. An example of a quadratic function with no real roots is given by,

Notice that the discriminant of f(x) is negative,

This function is graphically represented by a parabola that opens upward whose vertex lies above the x-axis. Thus, the graph can never intersect the x-axis and has no roots, as shown below,

Case 2: One Real Root

If the discriminant of a quadratic function is equal to zero, that function has exactly one real root and crosses the x-axis at a single point. To see this, we set b 2 &minus4ac = 0 in the quadratic formula to get,

Notice that is the x-coordinate of the vertex of a parabola. Thus, a parabola has exactly one real root when the vertex of the parabola lies right on the x-axis. The simplest example of a quadratic function that has only one real root is,

where the real root is x = 0.

Another example of a quadratic function with one real root is given by,

Notice that the discriminant of f(x) is zero,

This function is graphically represented by a parabola that opens downward and has vertex (3/2, 0), lying on the x-axis. Thus, the graph intersects the x-axis at exactly one point (i.e. has one root) as shown below,

Case 3: Two Real Roots

If the discriminant of a quadratic function is greater than zero, that function has two real roots (x-intercepts). Taking the square root of a positive real number is well defined, and the two roots are given by,

An example of a quadratic function with two real roots is given by,

Notice that the discriminant of f(x) is greater than zero,

This function is graphically represented by a parabola that opens upward whose vertex lies below the x-axis. Thus, the graph must intersect the x-axis in two places (i.e. has two roots) as shown below,

In the next section we will use the quadratic formula to solve quadratic equations.


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There are many laws of exponents that should be memorized and practiced in order to be thoroughly understood. The following exponent laws are detailed more thoroughly with examples on the exponential powers page and the radicals & roots page.

Exponential Equations & the Number of Solutions

One property of exponential equations that is initially confusing to some students is determining how many solutions an equation will have.

    Exponential equations with one term and an even power will have up to 2 solutions.

An even exponent hides the sign of its roots (e.g., x 2 =4 x = 2 and x = -2).

One of the more commonly tested properties of exponents and exponential equations is that an even exponent hides the sign of its roots. Consider the following example:

Techniques for Solving Exponential Equations

As noted above, an exponential equation has one or more terms with a base that is raised to a power that is not 1. While there is no formula for solving an exponential equation, the following examples provide some insight into common techniques used in finding the unknown value in an exponential equation.

Technique 1: Isolate and Raise to the Inverse Exponent

Arrange the term with an exponent on one side of the equation and the other terms on the other side of the equation. Raise both sides of the equation to the inverse exponent.

Work to isolate the x 4 term by subtracting 6 from both sides and then dividing both sides by 3.

In order to isolate x, since x is raised to the 4/1 power, raise both sides to the inverse power (i.e., 1/4).

Technique 2: Solve Through Factoring

Isolating an exponent often makes solving an equation easier.

For a more detailed explanation of this technique, please visit the factoring study guide and the quadratic equations study guide. Arrange all similar terms on one side of the equal sign and then factor.

Divide each term by 2, which is a common factor, and then subtract the number on the right side of the equation.

Using factoring rules, simplify and solve the exponential equation.


Watch the video: Section Transforming with Powers and Roots (June 2022).


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