Evaluate x. \left(\dfrac{x^{4}}{7^{-8}}\right)^{-7}

Simplify Fraction with Negative Exponent: We need to simplify the expression $\left(\dfrac{x^{4}}{7^{-8}}\right)^{-7}$. First, we will simplify the fraction inside the parentheses by dealing with the negative exponent on the denominator. \[ \left(\dfrac{x^{4}}{7^{-8}}\right)^{-7} = \left(x^{4} \cdot 7^{8}\right)^{-7} \]Apply Power of a Power Rule: Now, we apply the power of a power rule, which states that $(a^m)^n = a^{mn}$, to both $x^{4}$ and $7^{8}$ inside the parentheses. \[ \left(x^{4} \cdot 7^{8}\right)^{-7} = x^{4 \cdot (-7)} \cdot 7^{8 \cdot (-7)} \]Multiply Exponents: Next, we multiply the exponents to simplify the expression further. \[ x^{4 \cdot (-7)} \cdot 7^{8 \cdot (-7)} = x^{-28} \cdot 7^{-56} \]Rewrite with Positive Exponents: Now, we can rewrite the expression with positive exponents by taking the reciprocal of the base. \[ x^{-28} \cdot 7^{-56} = \dfrac{1}{x^{28}} \cdot \dfrac{1}{7^{56}} \]Combine Fractions: Finally, we combine the two fractions into one. \[ \dfrac{1}{x^{28}} \cdot \dfrac{1}{7^{56}} = \dfrac{1}{x^{28} \cdot 7^{56}} \]

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Evaluate $x$ . $\newline$ $\left(\dfrac{x^{4}}{7^{-8}}\right)^{-7}$

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\newline

\left(\dfrac{x^{4}}{7^{-8}}\right)^{-7}

Simplify Fraction with Negative Exponent: We need to simplify the expression $\left(\dfrac{x^{4}}{7^{-8}}\right)^{-7}$ . First, we will simplify the fraction inside the parentheses by dealing with the negative exponent on the denominator. $\newline$ $\left(\dfrac{x^{4}}{7^{-8}}\right)^{-7} = \left(x^{4} \cdot 7^{8}\right)^{-7}$
Apply Power of a Power Rule: Now, we apply the power of a power rule, which states that $(a^m)^n = a^{mn}$ , to both $x^{4}$ and $7^{8}$ inside the parentheses. $\newline$ $\left(x^{4} \cdot 7^{8}\right)^{-7} = x^{4 \cdot (-7)} \cdot 7^{8 \cdot (-7)}$
Multiply Exponents: Next, we multiply the exponents to simplify the expression further. $\newline$ $x^{4 \cdot (-7)} \cdot 7^{8 \cdot (-7)} = x^{-28} \cdot 7^{-56}$
Rewrite with Positive Exponents: Now, we can rewrite the expression with positive exponents by taking the reciprocal of the base. $\newline$ $x^{-28} \cdot 7^{-56} = \dfrac{1}{x^{28}} \cdot \dfrac{1}{7^{56}}$
Combine Fractions: Finally, we combine the two fractions into one. $\newline$ $\dfrac{1}{x^{28}} \cdot \dfrac{1}{7^{56}} = \dfrac{1}{x^{28} \cdot 7^{56}}$