Question

2 match each expression with its equivalent form using rational exponents. \\(\sqrt4{x^6}\\) \\(\sqrt5{x^{15}}\\) \\(\sqrt10{x^4}\\) \\(x^{\frac{3}{5}}\\) \\(x^{\frac{2}{5}}\\) \\(x^{\frac{3}{2}}\\)

Explanation:

Step1: Recall the formula for converting radicals to rational exponents: $\sqrt[n]{x^m} = x^{\frac{m}{n}}$

For $\sqrt[4]{x^6}$:

Using the formula, $n = 4$ and $m = 6$, so $\sqrt[4]{x^6}=x^{\frac{6}{4}} = x^{\frac{3}{2}}$

For $\sqrt[5]{x^{15}}$:

Using the formula, $n = 5$ and $m = 15$, so $\sqrt[5]{x^{15}}=x^{\frac{15}{5}} = x^{3}$ (Wait, maybe there's a typo in the target exponents, but assuming the target exponents are like $x^{\frac{3}{2}}$, $x^{3}$, $x^{\frac{2}{5}}$ (from $\sqrt[10]{x^4}=x^{\frac{4}{10}} = x^{\frac{2}{5}}$))

For $\sqrt[10]{x^4}$:

Using the formula, $n = 10$ and $m = 4$, so $\sqrt[10]{x^4}=x^{\frac{4}{10}} = x^{\frac{2}{5}}$

Step2: Match each radical to its rational exponent form

• $\sqrt[4]{x^6}$ matches $x^{\frac{3}{2}}$ (assuming one of the targets is $x^{\frac{3}{2}}$)
• $\sqrt[5]{x^{15}}$ matches $x^{3}$ (assuming one of the targets is $x^{3}$)
• $\sqrt[10]{x^4}$ matches $x^{\frac{2}{5}}$ (assuming one of the targets is $x^{\frac{2}{5}}$)

Answer:

• $\sqrt[4]{x^6}$ corresponds to $x^{\frac{3}{2}}$
• $\sqrt[5]{x^{15}}$ corresponds to $x^{3}$
• $\sqrt[10]{x^4}$ corresponds to $x^{\frac{2}{5}}$

(Note: The exact matching depends on the precise target exponents, but the above uses the radical - to - rational exponent conversion formula $\sqrt[n]{x^m}=x^{\frac{m}{n}}$)