Laws of Exponents

(i) multiplication of two powers:

Thus

2 ³ × 2 ⁵ = 2 ^{3 + 5} = 2 ⁸

In symbols, we can write that if c is any number, then

The illustration above which shows why multiplying two exponentials together gives a new exponential whose exponent is the sum of the original exponents can clearly be extended to products of three or more exponentials with the same base. The total number of factors in the product is equal to the sum of the factors from all exponentials involved, so the exponent in the simplified product will be just the sum of the exponents in the factors. This is illustrated in the second example below.

examples:

5 ⁷ · 5 ⁴ = 5 ^{7 + 4} = 5 ¹¹

7 ² · 7 ⁵ · 7 ³ · 7 ⁸ = 7 ^{2 + 5 + 3 + 8} = 7 ¹⁸

(ii) division of one power by another:

Here, the four factors of 2 in the denominator cancel four of the factors of 2 in the numerator, leaving a net of three factors of 2 in the numerator. The denominator of 1 can simply be dropped to get the final result 2 ³ overall. Notice that this simplification can be written more compactly as

since if we are counting up overall factors of 2 in the expression, the number of factors of 2 in the denominator must be subtracted from the number of factors of 2 in the numerator.

In symbols, if c is any nonzero number, and m is a larger number than n, we can write

Note that if we started with

then the denominator has more factors of 2 than does the numerator. When all possible cancellation of factors is done, there will be three factors of 2 left on the bottom, and none on the top:

So, in symbols, if c is any nonzero number, but now n > m, we get

examples:

From this last example, you can see that if two or more powers with the same base are multiplied in the numerator or the denominator or both, then the final result will have a power equal to the sum of all exponents in the numerator minus the sum of all exponents in the denominator. This only works for those powers that have the same base.

(iii) raising a power to a power:

This amounts to noting that

(5 ² ) ³ = 5 ^{2 × 3} = 5 ⁶

In symbols, if c is any number, then

In the last form in the box, we have used the algebraic convention that the product n × m can be written simply as nm.

example:

(3 ⁴ ) ² = 3 ^{4 × 2} = 3 ⁸ (whereas 3 ⁴ · 3 ² = 3 ^{4 + 2} = 3 ⁶ )

To summarize so far:

• When a power is raised to a power you multiply the two exponents together.
• When a power is multiplied by another power with the same base, you add the exponents.
• When a power is divided by another power with the same base, you subtract the second exponent from the first.

(iv) raising a product to a power:

In general, then, if c and d are any numbers,

example:

( 5 × 3 ) ⁷ = 5 ⁷ × 3 ⁷

(v) raising a quotient or a fraction to a power:

If c is any number, and d is any nonzero number, then

So, for example