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. Recall the property of exponents for negative powers: \(a^{-n} = \frac{1}{a^n}\), where \(a \neq 0\) and \(n\) is a positive integer.
2. Apply this property to the expression \((\frac{2}{3})^{-2}\), which becomes \(\frac{1}{(\frac{2}{3})^2}\).
3. Simplify the denominator by squaring the fraction: \((\frac{2}{3})^2 = \frac{2^2}{3^2} = \frac{4}{9}\).
4. Rewrite the expression as \(\frac{1}{\frac{4}{9}}\), which is equivalent to multiplying by the reciprocal: \$1 \times \frac{9}{4} = \frac{9}{4}$.
5. Compare this result to the right side of the original equation \((\frac{3}{2})^2 = \frac{3^2}{2^2} = \frac{9}{4}\). Since both sides simplify to \(\frac{9}{4}\), the original statement is true.

Key Concepts

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. For example, a^(-n) = 1/(a^n). This rule allows us to rewrite expressions with negative exponents into more manageable forms.

Reciprocal of a Fraction

The reciprocal of a fraction is obtained by swapping its numerator and denominator. For instance, the reciprocal of (2/3) is (3/2). This concept is essential when dealing with negative exponents applied to fractions.

Exponentiation of Fractions

Raising a fraction to a power means raising both the numerator and denominator to that power separately. For example, (a/b)^n = (a^n)/(b^n). This rule helps simplify expressions involving powers of fractions.