Evaluate each expression without using a calculator. 27^1/3

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Recognize that the expression \(27^{1/3}\) represents the cube root of 27, because an exponent of \(\frac{1}{3}\) means the cube root.

Rewrite the expression as \(\sqrt[3]{27}\) to make it clearer that we are looking for a number which, when multiplied by itself three times, equals 27.

Recall that 27 is a perfect cube since \(3 \times 3 \times 3 = 27\).

Identify that the cube root of 27 is 3 because \(3^3 = 27\).

Therefore, \(27^{1/3}\) simplifies to 3.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Exponents and Powers

Exponents represent repeated multiplication of a base number. For example, 27 can be written as 3^3 because 3 multiplied by itself three times equals 27. Understanding how to manipulate exponents is essential for simplifying expressions involving powers.

Fractional Exponents

A fractional exponent like 1/3 indicates a root; specifically, the denominator of the fraction is the root's degree. For instance, x^(1/3) means the cube root of x. This concept allows rewriting roots as exponents, facilitating easier evaluation and simplification.

Evaluating Cube Roots

The cube root of a number is the value that, when multiplied by itself three times, gives the original number. For example, the cube root of 27 is 3 because 3^3 = 27. Recognizing perfect cubes helps in evaluating expressions like 27^(1/3) without a calculator.

Evaluate each expression without using a calculator. 271/3