Multiple Choice
Determine if the given square root evaluates to a real number.
11. 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 nested roots using fractional exponents: \(\sqrt{243} = 243^{\frac{1}{2}}\), so the entire expression becomes \(\left(243^{\frac{1}{2}}\right)^{\frac{1}{5}}\).
Use the power of a power property by multiplying the exponents: \(243^{\frac{1}{2} \times \frac{1}{5}} = 243^{\frac{1}{10}}\).
Express 243 as a power of a prime number: since \$243 = 3^5$, substitute to get \(\left(3^5\right)^{\frac{1}{10}}\).
Again, apply the power of a power rule by multiplying exponents: \(3^{5 \times \frac{1}{10}} = 3^{\frac{1}{2}}\), which is the square root of 3.
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