Multiple Choice
Use the quotient rule to divide, then simplify.
11. 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: \(^4\sqrt{80y^5}\). This means the 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 \(^4\sqrt{a \times b} = ^4\sqrt{a} \times ^4\sqrt{b}\) to separate the fourth root: \(^4\sqrt{16} \times ^4\sqrt{5} \times ^4\sqrt{y^5}\).
Simplify each part: \(^4\sqrt{16}\) is the fourth root of \$16\(, which can be expressed as \)16^{1/4}$; \(^4\sqrt{y^5}\) can be rewritten using exponent rules as \(y^{5/4} = y^{1 + 1/4} = y \times y^{1/4}\).
Combine the simplified parts back together: multiply the simplified root of 16, the \(y\) term outside the root, and the remaining fourth roots of 5 and \(y^{1/4}\) to write the expression in simplest form.
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