Question

which radical expression is equivalent to $b^{\frac{1}{4}}$?
choose 1 answer:
a $\frac{1}{(\sqrt{b})^4}$
b $\frac{1}{\sqrt4{b}}$
c $\sqrt4{b}$
d $\frac{1}{\sqrt4{b}}$

Explanation:

Step1: Recall the exponent - radical relationship

The general rule for converting a fractional exponent to a radical is \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\) (when \(n>0\)). For the expression \(b^{\frac{1}{4}}\), we have \(m = 1\) and \(n=4\).

Step2: Apply the rule

Using the rule \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\), when \(a = b\), \(m = 1\) and \(n = 4\), we get \(b^{\frac{1}{4}}=\sqrt[4]{b^{1}}=\sqrt[4]{b}\).

Now let's analyze the other options:

• Option A: \(\frac{1}{(\sqrt{b})^{4}}=\frac{1}{b^{2}}\) (since \((\sqrt{b})^{4}=b^{2}\)), which is not equal to \(b^{\frac{1}{4}}\).
• Option B: \(\frac{1}{\sqrt[4]{b}}=b^{-\frac{1}{4}}\), which is the reciprocal of \(b^{\frac{1}{4}}\), not equal.
• Option D: \(\frac{1}{\sqrt[4]{b}}=b^{-\frac{1}{4}}\), same as option B, not equal.

Answer:

C. \(\sqrt[4]{b}\)