Multiple Choice
Determine if the given square root evaluates to a real number.
9. Roots, Radicals, and Complex Numbers
Introduction to Radical Expressions
Multiple Choice
Simplify the root.
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Verified step by step guidance1
Recognize that the expression involves the fifth root of the square root of 243, which can be written as \(\sqrt[5]{\sqrt{243}}\).
Rewrite the square root as an exponent: \(\sqrt{243} = 243^{\frac{1}{2}}\).
Substitute this back into the fifth root expression: \(\sqrt[5]{243^{\frac{1}{2}}}\).
Use the property of exponents that \(\sqrt[n]{a^m} = a^{\frac{m}{n}}\) to combine the roots: \(243^{\frac{1}{2} \times \frac{1}{5}} = 243^{\frac{1}{10}}\).
Express 243 as a power of a prime number (since \$243 = 3^5$), then rewrite the expression as \(\left(3^5\right)^{\frac{1}{10}}\) and simplify the exponent by multiplying: \(3^{5 \times \frac{1}{10}}\).
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