Algebra Know-It-ALL

112 Powers and Roots

The value keeps getting larger without limit, doubling every time. The sequence of values is said to diverge.

In this particular example, it “approaches infinity.” But if you start with 2 and raise it to integer powers

that get larger and larger negatively, this is what happens:

2 −^1 = 1/2

2 −^2 = 1 / (2 × 2)= 1/4

2 −^3 = 1 / (2 × 2 × 2)= 1/8

2 −^4 = 1 / (2 × 2 × 2 × 2)= 1/16

↓

and so on, forever

The value keeps getting smaller and smaller, becoming half its former size every time you decrease the

integer power by 1, but always remaining positive. This sequence of values is said to converge. In this case

it “approaches 0.”

Reciprocal-of-Integer Powers

Now that we’ve seen what happens when a number is raised to an integer power, let’s find out

what goes on when a number is raised to a power that is the reciprocal of an integer.

Integer roots are reciprocal-of-integer powers

Suppose we take some number or quantity a, and raise it to the power 1/p where p is a positive

integer. We write this as

a1/p

We can surround the exponent with parentheses for clarity. If we do that to the above expres-

sion, we get

a(1/p)

In this case, the parentheses are not technically necessary because the whole ratio is written as

a superscript anyway.

When we take a reciprocal-of-integer power of a quantity, the result is often called a root.

If you have a number and raise it to the power 1/p, it is the same thing as taking the pth root

of that number. If p is a positive integer, then the pth root of a quantity is something we must

multiply by itself p times in order to get that quantity.

The square root

If the general formulas above confuse you, it can help if we look at an example. We know that

52 = 5 × 5 = 25