Complicated Exponent Laws (Index Laws)
So far, we have considered situations where one particular exponent’s law was used for simplifying expressions with exponents (indices). However, in most practical situations, more than one law is needed to simplify the expression.
The following example simplifies expressions with exponents (indices) using several exponent laws.
Example 1
Write $64^{\frac{2}{3}}$ in simplest form.
\( \begin{align} \displaystyle
64^{\frac{2}{3}} &= (4^3)^{\frac{2}{3}} \\
&= 4^{3 \times \frac{2}{3}} \\
&= 4^2 \\
&= 16
\end{align} \)
When more than one exponent law (index law) is used to simplify an expression, the following steps can be taken.
$\textit{Step 1}$: If an expression contains brackets, expand them first.
$\textit{Step 2}$: If an expression is a fraction, simplify each numerator and denominator, then divide (simplify across then down).
$\textit{Step 3}$: Express the final answer with positive exponents (indices).
The following examples illustrate the use of exponent laws (index law) for the multiplication and division of fractions.
Example 2
Simplify $\dfrac{(2x^2y^3)^3 \times 3(xy^4)^2}{6x^4 \times 2xy^4}$.
\( \begin{align} \displaystyle
\dfrac{(2x^2y^3)^3 \times 3(xy^4)^2}{6x^4 \times 2xy^4} &= \dfrac{2^3x^6y^9 \times 3x^2y^8}{(6 \times 2) \times x^{4+1}y^4} \\
&= \dfrac{(8 \times 3) \times x^{6+2} \times y^{9+8}}{12x^5y^4} \\
&= \dfrac{24x^8y^{17}}{12x^5y^4} \\
&= \dfrac{24}{12} \times x^{8-5} \times y^{17-4} \\
&= 2x^3y^{13}
\end{align} \)
Example 3
Simplify $a^{-2}b^4 \times (a^3b^{-4})^{-1}$, leaving your answer with positive exponents.
\( \begin{align} \displaystyle
a^{-2}b^4 \times (a^3b^{-4})^{-1} &= a^{-2}b^4 \times a^{-3}b^4 \\
&= a^{-2-3}b^{4+4} \\
&= a^{-5}b^8 \\
&= \dfrac{b^8}{a^5}
\end{align} \)
Example 4
Simplify $\Bigg(\dfrac{a^{-\frac{1}{2}}b^{-1}}{3^{-1}b^2}\Bigg)^{-1} \div \Bigg(\dfrac{3a^{-\frac{3}{2}}b^2}{a^{\frac{3}{4}}b^{\frac{1}{2}}}\Bigg)^2$, leaving your answer with positive exponents.
\( \begin{align} \displaystyle
\Bigg(\dfrac{a^{-\frac{1}{2}}b^{-1}}{3^{-1}b^2}\Bigg)^{-1} \div \Bigg(\dfrac{3a^{-\frac{3}{2}}b^2}{a^{\frac{3}{4}}b^{\frac{1}{2}}}\Bigg)^2 &= \dfrac{a^{\frac{1}{2}}b}{3b^{-2}} \div \dfrac{3^2a^{-3}b^4}{a^{\frac{3}{2}}b} \\
&= \dfrac{a^{\frac{1}{2}}b}{3b^{-2}} \times \dfrac{a^{\frac{3}{2}}b}{3^2a^{-3}b^4} \\
&= \dfrac{a^2b^2}{27a^{-3}b^2} \\
&= \dfrac{a^5}{27}
\end{align} \)
Example 5
Simplify $\dfrac{3^n \times 6^{n+1} \times 12^{n-1}}{3^{2n} \times 8^n}$.
\( \begin{align} \displaystyle
\dfrac{3^n \times 6^{n+1} \times 12^{n-1}}{3^{2n} \times 8^n} &= \dfrac{3^n \times (3 \times 2)^{n+1} \times (2^2 \times 3)^{n-1}}{3^{2n} \times 2^{3n}} \\
&= \dfrac{3^n \times 3^{n+1} \times 2^{n+1} \times 2^{2n-2} \times 3^{n-1}}{3^{2n} \times 2^{3n}} \\
&= \dfrac{3^{3n} \times 2^{3n-1}}{3^{2n} \times 2^{3n}} \\
&= 3^n \times 2^{-1} \\
&= \dfrac{3^n}{2}
\end{align} \)
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