Express $(4 \frac{1}{5})^4$ in simplest radical form.

Steps to Solve

1. Convert the mixed fraction to an improper fraction

To convert the mixed fraction $4 \frac{1}{5}$ to an improper fraction, multiply the whole number part (4) by the denominator (5) and add the numerator (1). This result becomes the new numerator, and the denominator stays the same. $4 \frac{1}{5} = \frac{(4 \times 5) + 1}{5} = \frac{20 + 1}{5} = \frac{21}{5}$

2. Raise the improper fraction to the power of 4

Now, raise the improper fraction $\frac{21}{5}$ to the power of 4. Apply the exponent to both the numerator and the denominator. $(\frac{21}{5})^4 = \frac{21^4}{5^4} = \frac{194481}{625}$

3. Express as a mixed radical form

Since we are asked to express the answer in simplest radical form, we need to check if we can simplify the fraction by taking out any perfect powers. However, since the question specifically asks for the answer in simplest radical form and $21^4$ and $5^4$ are already perfect fourth powers (albeit of 21 and 5 respectively), we recognize that this means we should express the answer by interpreting the exponent in the expression $(4 \frac{1}{5})^4 $ as applying to the entire number rather than individual factors, implying we want to determine $\sqrt[4]{(4\frac{1}{5})^4}$. So we rewrite the given expression as an integer: $(\frac{21}{5})^4=(\frac{21}{5})\frac{21}{5}\frac{21}{5}*\frac{21}{5}$ Then in the simplest radical form we obtain $21/5$ $21/5 = 4\frac{1}{5}$