EDGE Math 8

Laws of Exponents

Consider the exponential expression ${(a^{3})}^{4} .$ This expression means

a^{3} \cdot a^{3} \cdot a^{3} \cdot a^{3} .

Since $a^{3} = a \cdot a \cdot a,$ the expression ${(a^{3})}^{4}$ indicates that you need to multiply 4 groups of the expression $a \cdot a \cdot a$ or 12 $a' s.$ Thus,

{(a^{3})}^{4} = a^{3 \times}^{4} = a^{12} .

This illustrates the second law of exponents.

Power Rule
If $a$ is a real number and $m$ and $n$ are positive integers, then

{(a^{m})}^{n} = a^{m n} .

This rule means that when you are raising an exponential expression to a power, you just need to multiply the exponents.

Example 1

Simplify each expression.

a. ${(2^{2})}^{3}$

b. ${(x^{6})}^{2}$

c. ${(p^{4})}^{4}$

Solution

a. ${(2^{2})}^{3} = 2^{2 \times 3} = 2^{6} = 64$

b. ${(x^{6})}^{2} = x^{6 \times 2} = x^{12}$

c. ${(p^{4})}^{4} = p^{4 \times 4} = p^{16}$