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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B
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Verified step by step guidance1
Recognize that the expression is a fourth root of the product \$256a^8b^9$, which can be written as \(\sqrt[4]{256a^8b^9}\).
Rewrite the fourth root of a product as the product of the fourth roots: \(\sqrt[4]{256} \times \sqrt[4]{a^8} \times \sqrt[4]{b^9}\).
Simplify each part separately:
- \(\sqrt[4]{256}\) is the fourth root of 256.
- \(\sqrt[4]{a^8}\) can be simplified using the property \(\sqrt[n]{x^m} = x^{m/n}\).
- \(\sqrt[4]{b^9}\) can also be simplified similarly.
Calculate the exponents after applying the root:
- For \(a^8\), the exponent becomes \$8/4\(.
- For \)b^9\(, the exponent becomes \)9/4$, which can be split into an integer part and a fractional part to separate the radical.
Combine the simplified terms, expressing any leftover fractional exponents as a fourth root, to write the expression in simplest form.
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