Evaluate the following exponents
Example 1. $6^4$
Solution
$6^4$ = 6 * 6 * 6 * 6 = 1296
Example 2. $(-7)^{-2}$
Solution
$\begin{align*}
\left (-7 \right )^{-2}
& = \frac{1}{(-7)^2}
\\
& = \frac{1}{49}
\\
\end{align*}
$
Example 3. $ (\frac{-3}{4})^{-3}$
Solution
$\begin{align*}
\left (\frac{-3}{4} \right )^{-3}
& = \left (\frac{4}{-3} \right )^3
\\
& = \left (\frac{-4}{3} \right )^3
\\
& = \frac{-4}{3} × \frac{-4}{3} × \frac{-4}{3}
\\
& = \frac{-64}{27}
\\
\end{align*}
$
Example 4. $6^4 × 6^2$
Solution
$6^4 × 6^2 = 6^{4 + 2} = 6^6 = 46656$
Example 5. $2^4 × 2^{-3}$
Solution
$2^4 × 2^{-3} = 2^{4 - 3} = 2^1 = 2$
Example 6. $(\frac{3}{4})^4 × (\frac{5}{4})^{-2}$
Solution
$\begin{align*}
\left (\frac{3}{4} \right )^4 × \left (\frac{5}{4} \right )^{-2}
& = \left (\frac{3}{4} \right )^4 × \left (\frac{4}{5} \right )^{2}
\\
& = \frac{3^4}{4^4} × \frac{4^2}{5^2}
\\
& = \frac{3^4}{4^2 × 5^2}
\\
& = \frac{81}{16 × 25}
\\
& = \frac{81}{400}
\end{align*}
$
Example 7. $(5^2)^3$
Solution $(5^2)^3 = 5^{2 * 3} = 5^6 = 15625$
Example 8. $\left ( \left [\frac{-5}{4} \right ]^2 \right )^{-3}$
Solution
$\begin{align*}
\left ( \left [\frac{-5}{4} \right ]^2 \right )^{-3}
& = \left [\frac{-5}{4} \right ]^{2 × ({-3})} \\
& = \left [\frac{-5}{4} \right ]^{-6} \\
& = \left [\frac{4}{-5} \right ]^{6} \\
& = \left [\frac{-4}{5} \right ]^{6} \\
& = \left [\frac{-4^6}{5^6} \right ] \\
& = \frac{-4096}{15625} \\
\end{align*}
$
Example 9. $ ( 3^{-1} + 6^{-1} ) ÷ \left (\frac{3}{4} \right )^{-1}$
Solution
$\begin{align*}
( 3^{-1} + 6^{-1} ) ÷ \left (\frac{3}{4} \right )^{-1}
& = \left ( \frac{1}{3} + \frac{1}{6} \right) ÷ \left (\frac{4}{3} \right )^{1} \\
& = \left ( \frac{2 + 1}{6} \right) ÷ \left (\frac{4}{3} \right ) \\
& = \left ( \frac{3}{6} \right) ÷ \left (\frac{4}{3} \right ) \\
& = \left ( \frac{1}{2} \right) ÷ \left (\frac{4}{3} \right ) \\
& = \left ( \frac{1}{2} × \frac{3}{4} \right) \\
& = \frac{3}{8} \\
\end{align*}
$
Example 10. If $ \left (\frac{3}{14} \right)^{-4} × \left (\frac{3}{14} \right) ^ {3x} = \left (\frac{3}{14} \right)^{5}$, then $x$ = ?
Solution
$\begin{align*}
\left (\frac{3}{14} \right)^{-4} × \left (\frac{3}{14} \right) ^ {3x}
& = \left (\frac{3}{14} \right)^{5} \\
& ⇒ -4 + 3x = 5 \\
& ⇒ 3x = 9 \\
& ⇒ x = 3 \\
\end{align*}
$
Example 11. By what number should $ \left (\frac{1}{2} \right)^{-1}$ be multiplied so that the product is $\left (\frac{-5}{4} \right )^{-1}$ ?
Solution
Let the required number be $x$. Then,
$\begin{align*}
\left (\frac{1}{2} \right)^{-1} × {x} = \left (\frac{-5}{4} \right )^{-1}
& ⇒ x = \frac {\left (\frac{-5}{4} \right )^{-1}} {\left (\frac{1}{2} \right)^{-1}} \\
& ⇒ x = \frac {\left (\frac{1}{2} \right)} {\left (\frac{-5}{4} \right )} \\
& ⇒ x = \left (\frac{1}{2} \right) × \left (\frac{-4}{5} \right ) \\
& ⇒ x = \frac{-4}{10} \\
& ⇒ x = \frac{-2}{5}
\end{align*}
$
Example 12. By what number should $ \left (\frac{-2}{3} \right)^{-3}$ be divided so that the quotient is $\left (\frac{4}{9} \right )^{-2}$ ?
Solution
Let the required number be $x$. Then,
$\begin{align*}
\frac {\left (\frac{-2}{3} \right)^{-3}} {x} = \left (\frac{4}{9} \right )^{-2}
& ⇒ \frac {\left (\frac{-2}{3} \right)^{-3}} {\left (\frac{4}{9} \right )^{-2}} = x \\
& ⇒ x = \frac {\left (\frac{4}{9} \right )^{2}} {\left (\frac{-2}{3} \right)^{3}} \\
& ⇒ x = \left (\frac{4}{9} \right )^{2} ÷ \left (\frac{-2}{3} \right)^{3} \\
& ⇒ x = \left (\frac{4}{9} \right )^{2} × \left (\frac{3}{-2} \right)^{3} \\
& ⇒ x = \left (\frac{4}{9} \right )^{2} × \left (\frac{-3}{2} \right)^{3} \\
& ⇒ x = \left (\frac{4 × 4 × -3 × -3 × -3}{9 × 9 × 2 × 2 × 2} \right ) \\
& ⇒ x = \left (\frac{4 × -3 }{9 × 2 } \right ) \\
& ⇒ x = \frac{-2}{3} \\
\end{align*}
$