Question

simplify $2,401^{-\frac{1}{4}}$. write your answer in simplest radical form.

Explanation:

Step1: Recall negative exponent rule

The negative exponent rule states that \( a^{-n}=\frac{1}{a^{n}} \). So, \( 2401^{-\frac{1}{4}}=\frac{1}{2401^{\frac{1}{4}}} \).

Step2: Recall fractional exponent rule

The fractional exponent rule is \( a^{\frac{m}{n}}=\sqrt[n]{a^{m}} \). For \( 2401^{\frac{1}{4}} \), this means \( \sqrt[4]{2401^{1}}=\sqrt[4]{2401} \).

Step3: Find the fourth root of 2401

We know that \( 7\times7\times7\times7 = 7^{4}=2401 \). So, \( \sqrt[4]{2401} = 7 \).

Step4: Substitute back

Since \( 2401^{\frac{1}{4}} = 7 \), then \( \frac{1}{2401^{\frac{1}{4}}}=\frac{1}{7} \).

Answer:

\(\frac{1}{7}\)