Length of edge = s

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Interior area = A

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A= s2 and 1/2 s=A = A

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Length of edge = s

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Figure 8-1 The area of a geometric square is equal to

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the 2nd power, or square, of the length of any edge Therefore, the length of any edge is equal to the 1/2 power, or square root, of the area

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114 Powers and Roots

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Length of edge = s

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Interior volume = V

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V = s3 and 1/3 s =V = V

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Length of edge = s

Length of edge = s

Figure 8-2 The volume of a geometric cube is equal to the

3rd power, or cube, of the length of any edge Therefore, the length of any edge is equal to the 1/3 power, or cube root, of the volume

The third power is often called the cube We can say, 4 cubed equals 64 Now if we go with the reciprocal power and work backwards, we get 641/3 = 4 This can be read as, The cube root of 64 equals 4 The 3rd power is called the cube and the 1/3 power is called the cube root because of the relationship between the edges and the interior volume of a geometric cube For any perfect cube, the volume is equal to the 3rd power of the length of any edge (Fig 8-2) Going the other way, the length of any edge is equal to the 1/3 power of the volume The figure also shows the radical notation for the cube root The fact that the radical refers to the cube root, rather than the square root, is indicated by the small numeral 3 in the upper-left part of the radical symbol

Higher roots When p is a positive integer equal to 4 or more, people write or talk about the numerical powers and roots directly That s because geometric hypercubes having 4 dimensions or more are not commonly named A 4-dimensional hypercube is technically called a tesseract, but you should expect incredulous stares from your listeners if you say 2 tesseracted is 16 or The tesseract root of 81 is 3 Here are some examples of higher powers and roots You can check the larger ones on your calculator if you like

24 = 16 34 = 81 so 161/4 = 2 so 811/4 = 4

Reciprocal-of-Integer Powers

56 = 15,625 so 15,6251/6 = 5 37 = 2,187 so ( 2,187)1/7 = 3 ( 5)9 = 1,953,125, so ( 1,953,125)1/9 = 5 64 = 1,296 so 1,2961/4 = 6 ( 6)4 = 1,296 so 1,2961/4 = 6 What The radical notation can be used for any integer root For the 1/n power, a small numeral n is placed in the upper left part of the radical symbol If you use this notation, you must be sure that the radical symbol extends completely over the quantity of which you want to take the root If you use the fractional notation, parentheses, brackets, and braces should be used to define the quantity of which you want to take the root

Are you confused

Now you will ask, Can 6 and 6 both be valid 4th roots of 1,296 The answer is Yes Both 6 and 6 will work here: 6 6 6 6 = 1,296 and ( 6) ( 6) ( 6) ( 6) = 1,296 If you multiply any negative number by itself an even number of times, you ll get a positive number Therefore, if you have some number a and its additive inverse a, and then you raise both of those numbers to an even positive integer power p, you will get ( a)p = a p every time! If we call ( a)p or a p by some other name such as b, then the pth root of b is ambiguous That would mean, for example, 161/4 = 2 and 2 811/4 = 3 and 3 15,6251/6 = 5 and 5 It could even mean something as simple, and yet as troubling, as 11/2 = 1 and 1 Mathematicians get around this problem by saying that whenever two numbers at once are the result of a reciprocal power, the positive value is the correct one, unless otherwise specified That means 161/4 = 2 811/4 = 3