Exponent Rules Solutions

Use the exponent rules to simplify an expression or expressions that are being multiplied together.
Exponents can be rational numbers $ℚ$ . Recall that integers are a subset of rational numbers.

Exponent Rules
Where ${a, b, c \in ℚ} and a > 0$

$a^{b} \cdot a^{c} = a^{b + c}$
${(a^{b})}^{c} = a^{b \cdot c}$
${(a b)}^{c} = a^{c} b^{c}$
$a^{b} = \frac{1}{a^{- b}} or a^{- b} = \frac{1}{a^{b}}$
$a^{0} = 1$
${(\frac{a}{b})}^{c} = \frac{a^{c}}{b^{c}}$
$\frac{a^{b}}{a^{c}} = a^{b - c} or \frac{1}{a^{c - b}}$
$a^{\frac{n}{d}} = \sqrt[d]{a^{n}}$

Example 1

Simplify.

${(2 x^{3} y^{4})}^{4}$

$2^{1 \cdot 4} x^{3 \cdot 4} y^{4 \cdot 4} 2^{4} x^{12} y^{16} 16 x^{12} y^{16}$

Example 2

Simplify the expression using only positive exponents.

$(\frac{x^{5} z^{3}}{x^{11} y^{- 6}}) {(\frac{z^{4}}{x^{10} y^{- 2}})}^{\frac{1}{2}} (\frac{y^{6} z^{3}}{x^{6}}) (\frac{z^{2}}{x^{5} y^{- 1}})$
$\frac{y^{6} z^{3 + 2}}{x^{6 + 5} y^{- 1}} = \frac{y^{6 + 1} z^{5}}{x^{11}} = \frac{y^{7} z^{5}}{x^{11}}$