Determine whether each statement is true or false. If false, correct the right side of the equation. (2/3)^-2 = (3/2)^2
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Identify the given equation: \( \left( \frac{2}{3} \right)^{-2} = \left( \frac{3}{2} \right)^2 \).
Recall the property of exponents: \( a^{-n} = \frac{1}{a^n} \).
Apply the property to \( \left( \frac{2}{3} \right)^{-2} \) to rewrite it as \( \left( \frac{3}{2} \right)^2 \).
Calculate \( \left( \frac{3}{2} \right)^2 \) by squaring both the numerator and the denominator.
Compare the results of both sides to determine if the statement is true or false.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Exponents and Negative Exponents
Exponents represent repeated multiplication of a base number. A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent. For example, a^(-n) = 1/(a^n). Understanding this concept is crucial for manipulating expressions involving negative exponents.
The reciprocal of a number is 1 divided by that number. For fractions, the reciprocal is obtained by swapping the numerator and denominator. This concept is essential when dealing with negative exponents, as it helps in rewriting expressions correctly and simplifying equations.
Squaring a fraction involves multiplying the fraction by itself. For example, (a/b)^2 = a^2/b^2. This concept is important for evaluating expressions and verifying the truth of equations involving fractions, especially when comparing two sides of an equation.