Multiple Choice
Use the quotient rule to divide, then simplify.
9. Roots, Radicals, and Complex Numbers
Simplifying Radical Expressions
Multiple Choice
Simplify the following.
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Verified step by step guidance1
Start with the expression: \(^4\sqrt{80y^5}\). Recognize that this is a fourth root of the product \$80y^5$.
Rewrite the radicand (the expression inside the root) by factoring it into prime factors and powers of variables: \(80 = 16 \times 5\), so \(80y^5 = 16 \times 5 \times y^5\).
Use the property of roots that \(\sqrt[n]{ab} = \sqrt[n]{a} \times \sqrt[n]{b}\) to separate the fourth root: \(^4\sqrt{16} \times ^4\sqrt{5} \times ^4\sqrt{y^5}\).
Simplify each part: \(^4\sqrt{16}\) simplifies because \$16 = 2^4\(, so \(^4\sqrt{16} = 2\). For \(^4\sqrt{y^5}\), rewrite the exponent as \)y^{4 + 1}$, which can be expressed as \(y^4 \times y^1\), so \(^4\sqrt{y^5} = y \times ^4\sqrt{y}\).
Combine all simplified parts: multiply the constants and variables outside the root, and keep the remaining terms inside the root. This results in \(2y \times ^4\sqrt{5y}\).
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