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
Recognize that the expression is a fourth root: \(^4\sqrt{256a^8b^9}\). This means we want to find the expression that, when raised to the 4th power, equals \$256a^8b^9$.
Rewrite the radicand (the expression inside the root) by factoring it into prime factors and powers: \(256 = 2^8\), so the expression becomes \(^4\sqrt{2^8 a^8 b^9}\).
Apply the property of radicals that \(^n\sqrt{x^m} = x^{m/n}\) to each factor inside the root separately: \(^4\sqrt{2^8} = 2^{8/4}\), \(^4\sqrt{a^8} = a^{8/4}\), and \(^4\sqrt{b^9} = b^{9/4}\).
Simplify the exponents where possible by dividing the powers: \(2^{8/4} = 2^2\), \(a^{8/4} = a^2\), and for \(b^{9/4}\), separate the integer and fractional parts as \(b^{8/4} \cdot b^{1/4} = b^2 \cdot b^{1/4}\).
Combine the simplified parts outside and inside the radical: multiply the integer powers outside the root to get \(2^2 a^2 b^2 = 4a^2b^2\), and write the remaining \(b^{1/4}\) as the fourth root of \(b\), so the final simplified form is \(4a^2b^2 \cdot {}^4\sqrt{b}\).
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