Determine whether each statement is true or false. If false, correct the right side of the equation. (2/3)^-2 = (3/2)²

검증된 단계별 안내

Identify the given equation: \(\left(\frac{2}{3}\right)^2 = \left(\frac{3}{2}\right)^2\).

Recall the property of exponents: when you square a fraction, you square both the numerator and the denominator separately. So, \(\left(\frac{a}{b}\right)^2 = \frac{a^2}{b^2}\).

Apply this property to both sides: the left side becomes \(\frac{2^2}{3^2} = \frac{4}{9}\), and the right side becomes \(\frac{3^2}{2^2} = \frac{9}{4}\).

Compare the two results: \(\frac{4}{9}\) is not equal to \(\frac{9}{4}\), so the original statement is false.

Correct the right side of the equation to match the left side: \(\left(\frac{2}{3}\right)^2 = \frac{4}{9}\).

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주요 개념

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

Properties of Exponents

Exponents indicate how many times a base is multiplied by itself. When raising a fraction to a power, both numerator and denominator are raised to that power separately, e.g., (a/b)^n = a^n / b^n. Understanding this helps evaluate expressions like (2/3)^2 correctly.

Reciprocals and Their Squares

The reciprocal of a fraction a/b is b/a. Squaring a fraction and squaring its reciprocal generally yield different results unless the fraction equals 1. For example, (2/3)^2 ≠ (3/2)^2, highlighting the importance of distinguishing between a fraction and its reciprocal.

Equality of Expressions

Two expressions are equal if they have the same value. To verify equality, simplify both sides fully. In this problem, comparing (2/3)^2 and (3/2)^2 requires calculating each side to check if they are equal or if one side needs correction.

Determine whether each statement is true or false. If false, correct the right side of the equation. (2/3)-2 = (3/2)2