EDGE Math 8

The laws of exponents can be extended to products involving more than two factors. For example,

a^{m} \cdot a^{n} \cdot a^{p} = a^{m + n + p}

and

{(a b c)}^{m} = a^{m} b^{m} c^{m} .

Exponential expressions can also be simplified using one or more of the laws of positive integer exponents.

Example

Simplify each expression.

1. $(2 a^{2} b) (3 a^{3} b^{4})$

2. ${(5 x^{4} y^{2})}^{2}$

3. ${(\frac{c^{6}}{2 d^{3}})}^{4}$ , where $d \neq 0$

Solution

1.	Use the associative and commutative properties of multiplication to rearrange the factors. Then apply the Product Rule.
	$(2 a^{2} b) (3 a^{3} b^{4}) = (2) (3) (a^{2} a^{3}) (b b^{4})$ $= 6 a^{5} b^{5}$

2.	Apply the Rule for a Product Raised to a Power and the Power Rule.
	${(5 x^{4} y^{2})}^{2} = 5^{2} {(x^{4})}^{2} {(y^{2})}^{2} = 25 x^{8} y^{4}$

3.	Apply the Rule for a Quotient Raised to a Power and the Power Rule.
	${(\frac{c^{6}}{2 d^{3}})}^{4} = \frac{{(c^{6})}^{4}}{{(2 d^{3})}^{4}} = \frac{c^{24}}{2^{4} {(d^{3})}^{4}}$ $= \frac{c^{24}}{16 d^{12}}$