Question

match each expression with its equivalent expression.
a. $sqrt4{x^5}$
b. $sqrt{x}$
c. $sqrt5{x^4}$
d. $sqrt5{x}$

$x^{\frac{5}{4}}$ $x^{\frac{1}{2}}$ $x^{\frac{1}{5}}$ $x^{\frac{4}{5}}$

drag each of the exponents given above into the appropriate area below, matching the expressions given in the problem statement.
a b c d

Explanation:

Step1: Recall the formula for converting radicals to exponents

The formula is \(\sqrt[n]{a^m}=a^{\frac{m}{n}}\).

Step2: Analyze Expression A (\(\sqrt[6]{x^5}\))

Using the formula, for \(\sqrt[6]{x^5}\), \(n = 6\) and \(m = 5\), so it is equivalent to \(x^{\frac{5}{6}}\).

Step3: Analyze Expression B (\(\sqrt{x}\))

\(\sqrt{x}=\sqrt[2]{x^1}\), so \(n = 2\) and \(m = 1\), which is equivalent to \(x^{\frac{1}{2}}\).

Step4: Analyze Expression C (\(\sqrt[5]{x^4}\))

For \(\sqrt[5]{x^4}\), \(n = 5\) and \(m = 4\), so it is equivalent to \(x^{\frac{4}{5}}\).

Step5: Analyze Expression D (\(\sqrt[5]{x}\))

\(\sqrt[5]{x}=\sqrt[5]{x^1}\), so \(n = 5\) and \(m = 1\), which is equivalent to \(x^{\frac{1}{5}}\).

Answer:

• A: \(x^{\frac{5}{6}}\)
• B: \(x^{\frac{1}{2}}\)
• C: \(x^{\frac{4}{5}}\)
• D: \(x^{\frac{1}{5}}\)