Questions: Simplify the exponential expression. [ left(frac7 x^4yright)^-3 ] (left(frac7 x^4yright)^-3=) (Simplify your answer. Use positive exponents only.)

$Simplify the exponential expression. [ left(frac7 x^4yright)^-3 ] (left(frac7 x^4yright)^-3=) (Simplify your answer. Use positive exponents only.)$

Transcript text: Simplify the exponential expression. \[ \left(\frac{7 x^{4}}{y}\right)^{-3} \] $\left(\frac{7 x^{4}}{y}\right)^{-3}=$ $\square$ (Simplify your answer. Use positive expenents only.)

Solution

Solution Steps

Step 1: Apply the Negative Exponent Rule

The expression $\left(\frac{7 x^{4}}{y}\right)^{-3}$ can be rewritten using the negative exponent rule, which states that $a^{-n} = \frac{1}{a^n}$. Applying this rule: \[ \left(\frac{7 x^{4}}{y}\right)^{-3} = \frac{1}{\left(\frac{7 x^{4}}{y}\right)^{3}}. \]

Step 2: Apply the Power to the Numerator and Denominator

Next, apply the exponent $3$ to both the numerator and the denominator inside the parentheses: \[ \frac{1}{\left(\frac{7 x^{4}}{y}\right)^{3}} = \frac{1}{\frac{(7 x^{4})^{3}}{y^{3}}}. \]

Step 3: Simplify the Numerator and Denominator

Simplify $(7 x^{4})^{3}$ by applying the power to each factor: \[ (7 x^{4})^{3} = 7^{3} \cdot (x^{4})^{3} = 343 x^{12}. \] Thus, the expression becomes: \[ \frac{1}{\frac{343 x^{12}}{y^{3}}}. \]

Step 4: Simplify the Complex Fraction

To simplify the complex fraction, multiply by the reciprocal of the denominator: \[ \frac{1}{\frac{343 x^{12}}{y^{3}}} = \frac{y^{3}}{343 x^{12}}. \]

Step 5: Final Simplified Form

The expression is now simplified with positive exponents: \[ \frac{y^{3}}{343 x^{12}}. \]