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What happens when we divide powers? Let's analyze this by dividing various powers of 2.

Try to make a conjecture about dividing powers that have the same base.

In both cases, the result is the original base raised to the power given by the difference of the two exponents.

For ,

and for .

The table on the left shows what happens when a power of 2 is divided by 2 1 . The table on the right shows what happens when a power of 2 is divided by 2 2

In the table on the left, notice the powers that result. Each resulting power can be found by subtracting 1 from the exponent of the original power. In the table on the right, each resulting power can be found by subtracting 2 from the exponent of the original power. This agrees with our original observation that when we divide two powers with the same base, the exponent of the resulting power is the difference of the exponents of the two dividends.

Consider another case. Let's divide a 4 by a 2. To do this, expand a 4 into a · a · a · a and a 2 into a · a . Next place them into the fraction

We can now cancel two a's from both numerator and denominator.

Cancellation is a shorthand process involving the properties of fractions. Also, point out that any number raised to the first power is that number itself.

After canceling, we find that .

In general, when we write the quotient and expand it into products of a's, the result is

which shows why this fact is valid.

Key Idea

When we divide a power of a by another smaller power of a , the result is a power of a , in which the exponent is the difference of the exponents of the two dividends. In symbols,

This holds true for any nonzero number a and whole numbers b and c.