116 Powers and Roots

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15,6251/6 = 5 11/2 = 1 You can indicate that you want to use the negative value by placing minus signs like this: (161/4) = 2 (811/4) = 3 (15,6251/6) = 5 (11/2) = 1 Sometimes you will actually want to let either the positive or the negative value be used In cases of that sort, you should throw a plus-or-minus sign ( ) into the mix, like this: (161/4) = 2 (811/4) = 3 (15,6251/6) = 5 (11/2) = 1 Here s another possible confusion-maker Always pay special attention to where the parentheses are placed if you see a negative number raised to a power Also, be careful if there are no parentheses at all If there s any doubt, it s best to place extra parentheses in an expression so everyone knows exactly what it means For example, ( 2)3 = ( 2) ( 2) ( 2) = 8 and 23 = (23) = (2 2 2) = 8 In contrast to this, ( 2)4 = ( 2) ( 2) ( 2) ( 2) = 16 but 24 = (24) = (2 2 2 2) = 16

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Negative reciprocal powers We still have not explored what happens when we raise a number to a negative reciprocal-ofinteger power You can probably figure out the meaning of an expression such as 125 1/3, or

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Multiplying and Dividing with Exponents

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125 to the 1/3 power You take the 1/3 power of 125, which is 5, and then take the reciprocal of that, which is 1/5 Mathematically, it goes like this: 125 1/3 = 125 (1/3) = 1/(1251/3) = 1/5

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Even roots of negative numbers What happens when you take an even root of a negative number The simplest example of this sort of problem is the square root of 1, but there are plenty of others What can you multiply by itself to get 1 Nothing that we ve defined yet! What is the 1/4 power of 16 Again, nothing we know of so far Mathematicians have defined quantities like this We will explore them in Chap 21 They re called imaginary numbers They have some fascinating properties Unlike division by 0 or the 0th root of 0, even roots of negative numbers can be tamed They are commonly used in science and engineering

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Here s a challenge!

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State the rule for negative reciprocal powers in general terms, where a is the base (the number to be raised to the power) and p is a positive integer

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The power to which we want to raise the base is 1/p, where p is some positive integer (We know that 1/p will be negative, because a negative divided by a positive always gives us a negative) If we use the method from the above example where we evaluated 125 1/3, then we have a 1/p = a (1/p) = 1/(a1/p)

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Multiplying and Dividing with Exponents

When we have a certain base number raised to two different powers, we get two different quantities But the fact that those quantities have the same base lets us take shortcuts in multiplication and division

Multiply by adding Let s state the general case first, and then check out an example Imagine a number and call it a This can be any number except 0 It doesn t have to be an integer or even a rational number Now imagine the quantities a m and a n, where m and n are integers If you multiply these two quantities, you get the same result as if you add m to n, and then raise the base a to that power We can write this as an equation:

a ma n = a(m+n)