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

Verified step by step guidance

1. Identify the given equation: \(\left(\frac{2}{3}\right)^2 = \left(\frac{3}{2}\right)^2\).
2. Recall the property of exponents: when you square a fraction, you square both the numerator and the denominator separately. So, \(\left(\frac{a}{b}\right)^2 = \frac{a^2}{b^2}\).
3. Apply this property to both sides: the left side becomes \(\frac{2^2}{3^2} = \frac{4}{9}\), and the right side becomes \(\frac{3^2}{2^2} = \frac{9}{4}\).
4. Compare the two results: \(\frac{4}{9}\) is not equal to \(\frac{9}{4}\), so the original statement is false.
5. Correct the right side of the equation to match the left side: \(\left(\frac{2}{3}\right)^2 = \frac{4}{9}\).

Key Concepts

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.