Write 27^2/3 in radical form and evaluate.

검증된 단계별 안내

Recognize that the expression \(27^{2/3}\) is written in exponential form with a rational exponent, where the numerator (2) is the power and the denominator (3) is the root.

Rewrite the expression using the property of rational exponents: \(a^{m/n} = \left( \sqrt[n]{a} \right)^m\). So, \(27^{2/3} = \left( \sqrt[3]{27} \right)^2\).

Evaluate the cube root \(\sqrt[3]{27}\) by finding the number that, when multiplied by itself three times, equals 27.

After finding the cube root, raise that result to the power of 2, which means squaring the number.

Express the final answer as the square of the cube root of 27, completing the evaluation step without calculating the numeric value.

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주요 개념

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Rational Exponents

Rational exponents express roots and powers simultaneously, where the numerator indicates the power and the denominator indicates the root. For example, a^(m/n) means the nth root of a raised to the mth power, allowing expressions like 27^(2/3) to be rewritten using radicals.

Radical Form Conversion

Converting expressions with rational exponents to radical form involves rewriting a^(m/n) as the nth root of a raised to the mth power, i.e., (√[n]{a})^m. This helps visualize and simplify expressions by using roots and powers explicitly.

Evaluating Radicals

Evaluating radicals requires finding the root of a number and then raising it to a power if needed. For example, to evaluate 27^(2/3), first find the cube root of 27, which is 3, then square it to get 9. This stepwise approach simplifies complex expressions.

Write 272/3 in radical form and evaluate.