Question

\\\sqrt6{x^4} \cdot \sqrt3{x^6}\\

Explanation:

⚡ Using what you learned: Integer and Rational Exponents

Step 1: Convert radicals to rational exponents

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

$$ \sqrt[6]{x^4} = x^{\frac{4}{6}} = x^{\frac{2}{3}} $$

$$ \sqrt[3]{x^6} = x^{\frac{6}{3}} = x^2 $$

Step 2: Multiply the terms

Using the product rule \( x^a \cdot x^b = x^{a+b} \):

$$ x^{\frac{2}{3}} \cdot x^2 = x^{\frac{2}{3} + 2} = x^{\frac{2}{3} + \frac{6}{3}} = x^{\frac{8}{3}} $$

Step 3: Convert back to radical form

Using \( x^{\frac{m}{n}} = \sqrt[n]{x^m} \):

$$ x^{\frac{8}{3}} = \sqrt[3]{x^8} = \sqrt[3]{x^6 \cdot x^2} = x^2 \sqrt[3]{x^2} $$

Answer:

$$ x^2 \sqrt[3]{x^2} \quad \text{or} \quad x^{\frac{8}{3}} $$