Question

\\ (5x)^{\frac{-5}{4}} = \\

Explanation:

⚡ Using what you learned: Expressions with Negative Exponents · Integer and Rational Exponents

Step 1: Apply the negative exponent rule

Using the rule \( a^{-n} = \frac{1}{a^n} \):

$$ (5x)^{-\frac{5}{4}} = \frac{1}{(5x)^{\frac{5}{4}}} $$

Step 2: Convert to radical form

Using the rational exponent rule \( a^{\frac{m}{n}} = \sqrt[n]{a^m} \):

$$ \frac{1}{(5x)^{\frac{5}{4}}} = \frac{1}{\sqrt[4]{(5x)^5}} $$

Step 3: Simplify the radical

Simplify the expression inside the radical:

$$ \sqrt[4]{(5x)^5} = \sqrt[4]{(5x)^4 \cdot (5x)} = 5x \sqrt[4]{5x} $$

Substitute this back into the fraction:

$$ \frac{1}{5x \sqrt[4]{5x}} $$

Step 4: Rationalize the denominator

Multiply the numerator and the denominator by \(\sqrt[4]{(5x)^3}\) to eliminate the radical in the denominator:

$$ \frac{1 \cdot \sqrt[4]{(5x)^3}}{5x \sqrt[4]{5x} \cdot \sqrt[4]{(5x)^3}} = \frac{\sqrt[4]{125x^3}}{5x \cdot 5x} = \frac{\sqrt[4]{125x^3}}{25x^2} $$

Answer:

$$ \frac{\sqrt[4]{125x^3}}{25x^2} $$