Question

which of the following is equivalent to \\(\sqrt4{a}\\)?

a.) \\(a^4\\)

b.) \\(a^{\frac{4}{2}}\\)

c.) \\(a^{\frac{1}{2}}\\)

d.) \\(a^{\frac{1}{4}}\\)

Explanation:

Identify the given expression

The given expression is:

$$ \sqrt[4]{a} $$

Apply the definition of rational exponents

The general rule converting a radical expression to an exponential expression is:

$$ \sqrt[n]{x^m} = x^{\frac{m}{n}} $$

For the given expression, the index of the radical is \(n = 4\) and the exponent of the radicand \(a\) is \(m = 1\).

Convert to exponential form

Applying the rule yields:

$$ \sqrt[4]{a} = a^{\frac{1}{4}} $$

This matches option (d).

Answer:

• a) \(a^4\)
• b) \(a^{\frac{4}{2}}\)

<mcq-option>c) \(a^{\frac{1}{2}}\)</mcq-correct>

• d) \(a^{\frac{1}{4}}\) (Correct answer)