Question

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

Explanation:

⚡ Using what you learned: Integer and Rational Exponents

Step 1: Apply negative exponent rule

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

Step 2: Convert to radical form

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

Step 3: Simplify the radical

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

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

Step 4: Rationalize the denominator

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

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

Answer:

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