Gaining full control of your powers: The Index Laws.

How to work with powers of numbers

You should be happy with squaring and cubing numbers, but what else can we do with powers?

Think of 23 x 22 => (2 x 2 x 2) x (2 x 2) => 25. All we've ended up doing is adding the powers together.

Similarly, if we divided they'd cancel out and we'd essentially be subtracting the powers.

Finally, (23)2 would give us 23 x 23 => (2 x 2 x 2) x (2 x 2 x 2) => 26. Here all that's happened is that we've multiplied the powers together!

This idea works for any numbers, not just 2, as long as you are working with powers that have the same base numbers. For example, it will work with 127 x 124 but not 132 x 93.

This gives us these three basic laws:

  • When multiplying powers, add the indices.
  • When dividing powers, subtract the indices
  • Doing powers of powers? Multiply the indices.

Taking it further

Imagine you are starting with 103 = 1000. Reducing the power by one gives 102 = 100. We've divided by 10 as we're multiplying by one less 10 to get our answer. Carrying this pattern on gives two more important properties of indices:

  • Anything to the power of 0 = 1
  • Negative powers are one over the positive power.

Again, these laws work with any numbers (including algebra!)

Also, considering the first law again, what if we had fractional indices?

31/2 x 31/2 would equal 3(1/2 + 1/2), which is 31 or just 3. This means that a power of 1/2 must be a square root - we need two numbers the same that multiply to make 3...

Similarly, a power of 1/3 is a cube root, as 41/3 x 41/3 x 41/3 = 41. And so on!

Meaning the essential Index Laws are:

  • Multiplying powers of the same number - add the indices
  • Dividing powers of the same number - subtract the indices
  • Raising a power to a power - multiply the indices
  • Anything to the power of zero = one
  • Negative power means one over the positive power
  • Fractional powers are roots

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