What Is the Connection Between Powers and Roots?

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A power:

ab

consists of a base a and an exponent b. The base a is the number that is to be multiplied with itself b times.

ab = a a aa b times ab ab abab b times = (ab)b = a

Rule

The Power Rules

a0 = 1 (ap) q = apq 1 aq = aq ap q = (ap) q ap aq = ap+q ap q = (aq)p ap aq = apq aqbq = abq (a b)p = ap bp aq bq = a bq (a b)p = ap bp

a0 = 1 (ap) q = apq 1 aq = aq ap q = (ap) q ap aq = ap+q ap q = (aq)p ap aq = apq aqbq = abq (a b)p = ap bp aq bq = a bq (a b)p = ap bp

Note! 1 a = a1

Pay special attention to the fact that a = a1 2 . This is called the square root of a. It then follows that an = a1 n. This is called the nth root of a.

If n is an even number, you get an even root. Some examples include a4,a6,a8,

In the same way, you get odd roots when n is an odd number. They look like this: a3,a5,a7,

The difference between even and odd roots is that even roots are only defined for a 0. In odd roots, a can be either positive or negative.

Note! You can not take the even root of a negative number. 24 = is undefined!

Example 1

Write a023b a2b2 as simply as possible

a023b a2b2 = 1 8 b1+2 a2 = 8b3 a2

Example 2

Write 32 xy4 x4y2 as simply as possible

32 xy4 x4y2 = 9x1+4 y2+4 = 9x5 y2

Example 3

Write 2a4 (ab)4 b5 (a2b)6 2b5 as simply as possible

2a4 (ab)4 b5 (a2b)6 2b5 = 2a4a4b4b5 a12b62b5 = 211a44+12b4+6+55 = 20a12b2 = a12b2

2a4 (ab)4 b5 (a2b)6 2b5 = 2a4a4b4b5 a12b62b5 = 211a44+12b4+6+55 = 20a12b2 = a12b2

Example 4

Write 2a3 2b 2a3 2 as simply as possible

2a3 2 b 2a3 2 = 2a3 2+3 2 b 2 = a6 2 b = a3b

Note! When you have a negative exponent, it’s usually best to write the power as a fraction. The reason to do this is that in general, positive exponents are much easier to deal with than negative ones.

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