Simplify the following.
${}^4\sqrt{80y^5}$

• A. $2y{.}^4\sqrt{5}$
• B. $16y{.}^4\sqrt{5y}$
• C. $16y$
• D. $2y.^4\(\sqrt{5y}\)$

Answer: (D)

Verified step by step guidance

1. Start with the expression \(^4\sqrt{80y^5}\), which means the fourth root of \$80y^5$.
2. Rewrite the radicand (the expression inside the root) by factoring it into prime factors and powers of variables: \(80 = 16 \times 5\), so \(80y^5 = 16 \times 5 \times y^5\).
3. Use the property of roots that \(\sqrt[n]{ab} = \sqrt[n]{a} \times \sqrt[n]{b}\) to separate the fourth root: \(^4\sqrt{16} \times ^4\sqrt{5} \times ^4\sqrt{y^5}\).
4. Simplify each part: \(^4\sqrt{16}\) is a perfect fourth power, so simplify it to 2; for \(^4\sqrt{y^5}\), rewrite the exponent as \(y^{4+1}\) and use the property \(^4\sqrt{y^4} = y\) times \(^4\sqrt{y}\).
5. Combine the simplified parts back together: multiply the constants and variables outside the root, and keep the remaining factors inside the fourth root to write the expression in simplest form.