Fractional Exponent Master Class

Part 1: Discovering the Rule

Before we touch fractions, let's look at regular numbers.

Look at this multiplication:

                $2^3 \times 2^2$
            

Expand it: $(2 \cdot 2 \cdot 2) \times (2 \cdot 2)$

How many 2s are there in total?

You found there are 5 twos. So $2^3 \times 2^2 = 2^5$.

Notice that $3 + 2 = 5$.

The Rule: When multiplying the same base, we ADD the powers.

Let's use that rule for $9^{1/2}$.

$$ 9^{1/2} \times 9^{1/2} = 9^{(1/2 + 1/2)} = 9^1 = 9 $$

This means: "Multiplying $9^{1/2}$ by itself gives 9."

Question: What number, times itself, equals 9?

If $9^{1/2} = 3$, and $\sqrt{9} = 3$, then they are the same thing!

$$ x^{\frac{1}{n}} = \sqrt[n]{x} $$