Simplify the expression. [tex]\left(\frac{4}{3}\right)^{-2}=\frac{3^2}{4^2}[/tex]
Answer (1)
Apply the property of negative exponents to rewrite the expression: ( 3 4 ) − 2 = ( 4 3 ) 2 .
Apply the power to both numerator and denominator: ( 4 3 ) 2 = 4 2 3 2 .
Calculate the squares: 4 2 3 2 = 16 9 .
The simplified expression is 16 9 .
Explanation
Understanding the Problem We are asked to simplify the expression ( 3 4 ) − 2 . The expression is claimed to be equal to 4 2 3 2 which is stated to be not a valid fraction. We need to simplify the given expression and determine if the provided simplification is correct.
Using the Negative Exponent Property To simplify the expression, we will use the property that a fraction raised to a negative power is equal to the reciprocal of the fraction raised to the positive power. In other words, ( b a ) − n = ( a b ) n . Applying this property to our expression, we get ( 3 4 ) − 2 = ( 4 3 ) 2 .
Applying the Power to Numerator and Denominator Next, we use the property that a fraction raised to a power is equal to the numerator raised to that power divided by the denominator raised to that power. In other words, ( b a ) n = b n a n . Applying this property, we get ( 4 3 ) 2 = 4 2 3 2 .
Calculating the Squares Now, we calculate 3 2 and 4 2 . We have 3 2 = 3 × 3 = 9 and 4 2 = 4 × 4 = 16 . Therefore, 4 2 3 2 = 16 9 .
Final Answer Thus, the simplified expression is 16 9 . The original expression ( 3 4 ) − 2 simplifies to 16 9 . The initial claim that 4 2 3 2 is not a valid fraction is incorrect, as 16 9 is a valid fraction. Therefore, the simplified expression is 16 9 .
Examples
Fractions and exponents are used in many real-world applications, such as calculating growth rates, scaling recipes, and determining financial returns. For example, if you invest money at a certain interest rate compounded annually, the formula for the future value of your investment involves raising a fraction (1 + interest rate) to the power of the number of years. Understanding how to simplify expressions with fractions and exponents is essential for making informed decisions in various fields.
