Study Guide

Basic Operations - Powers

Powers

As young math whipper-snappers, we added five 4s like this:

4 + 4 + 4 + 4 + 4

Then we learned a shorter way to write it, like this:

4 × 5

Repeated addition can be written as multiplication.

Now that we're all grown up in math, we've got a shorter way to write repeated multiplication, too. If we want to multiply 5 four times, we can write 5 × 5 × 5 × 5. But a shorter way is to use powers.

Powers, also known as exponents, are a short way to write long strings of multiplication.

Instead of writing 5 × 5 × 5 × 5, write 54 instead, which means "multiply four 5s together." We pronounce 54 as "5 to the fourth power."

Or how about this for a timesaver: isn't 109 easier to write than 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10?

Grab a magnifying glass and let's look closer at 109. The big number, 10, is the base. This is the number that's being multiplied a lot of times. The smaller number hiding in the upper right-hand corner is the exponent. It's a little shy, but it tells us how many of the base we're multiplying. The base is 10 and the exponent is 9, so we're multiplying nine 10s together. And that's one big number. Which is exactly what exponents are most useful for: writing very big numbers.

Important Rules of Powers

  1. Any number to the power of 0 is 1.
    • 25^0 = 1
    • 0.02764187^0=1
  2. Any number to the power of 1 is itself.
    • 5^1= 5
    • -(218)^1 = -218
  3. When you see a number raised to a negative exponent, take the reciprocal of the number (flip the fraction) and then change the exponent from negative to positive.
    • 2^-3 - (1 / 2)^3 - 1 / 8
  4. Remember to follow PEMDAS, the order of operations. For example, -6^2 and (-6)^2 are two different problems. 
    • The first, -6^2 , means to take the negative of 6^2. The answer is the negative of 36 or -36
    • The second problem, (-6)^2, means to square -6, so it would simplify as -6 x -6, or 36.

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