Simplify each expression. Write answers without negative exponents. Assume all variables represent positive real numbers. 8^2/3

검증된 단계별 안내

Recognize that the expression is an exponentiation of a number: \(8^{2/3}\). This means you are raising 8 to the power of two-thirds.

Recall the rule for fractional exponents: \(a^{m/n} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}\). Here, \(m=2\) and \(n=3\).

Apply the cube root first: find \(\sqrt[3]{8}\), which means the number that when cubed gives 8.

After finding \(\sqrt[3]{8}\), raise that result to the power of 2, i.e., square it.

Write the final expression without any negative exponents, ensuring the answer is simplified completely.

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주요 개념

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

Exponents and Powers

Exponents indicate how many times a base is multiplied by itself. For example, 8^2 means 8 multiplied by itself twice. Understanding how to manipulate exponents is essential for simplifying expressions involving powers.

Rational Exponents

Rational exponents like 8^(2/3) represent roots and powers combined. The denominator (3) indicates the root (cube root), and the numerator (2) indicates the power. So, 8^(2/3) means the cube root of 8 squared.

Simplifying Expressions Without Negative Exponents

Negative exponents represent reciprocals, but the problem requires answers without them. This means rewriting expressions so all exponents are positive, often by using properties like a^(-n) = 1/a^n.

Simplify each expression. Write answers without negative exponents. Assume all variables represent positive real numbers. 82/3