Question

write the following in simplified radical form.
\sqrt4{y^{17}}
assume that the variable represents a positive real number.

Explanation:

Step1: Divide exponent by index

We have the fourth root of \( y^{17} \). To simplify, we divide the exponent \( 17 \) by the index \( 4 \). Using division, \( 17\div4 = 4 \) with a remainder of \( 1 \) (since \( 4\times4 = 16 \) and \( 17-16 = 1 \)).

Step2: Rewrite the radical

Using the property of radicals \( \sqrt[n]{a^{m}}=a^{\frac{m}{n}} \) and also \( \sqrt[n]{a^{m}\cdot b}=\sqrt[n]{a^{m}}\cdot\sqrt[n]{b} \), we can rewrite \( \sqrt[4]{y^{17}} \) as \( \sqrt[4]{y^{16}\cdot y} \).

Step3: Simplify the perfect fourth power

Since \( \sqrt[4]{y^{16}} = y^{4} \) (because \( (y^{4})^{4}=y^{16} \)), we can simplify \( \sqrt[4]{y^{16}\cdot y} \) as \( y^{4}\sqrt[4]{y} \).

Answer:

\( y^{4}\sqrt[4]{y} \)