Simplify $\left(\frac{16}{81}\right)^{-\frac{3}{4}}$
Answer (2)
The expression ( 81 16 ) − 4 3 simplifies to 8 27 by first flipping the fraction due to the negative exponent, then expressing it as powers and simplifying. Each step uses properties of exponents and fraction operations. Ultimately, the final answer is 8 27 .
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Rewrite the expression using the property of negative exponents: ( 81 16 ) − 4 3 = ( 16 81 ) 4 3 .
Express the fraction as a power: 16 81 = ( 2 3 ) 4 .
Simplify the exponent using the property ( a m ) n = a mn : ( ( 2 3 ) 4 ) 4 3 = ( 2 3 ) 3 .
Calculate the final result: ( 2 3 ) 3 = 8 27 . The simplified expression is 8 27 .
Explanation
Understanding the Problem We are asked to simplify the expression ( 81 16 ) − 4 3 . This involves dealing with a fraction raised to a negative fractional power. Let's break it down step by step.
Flipping the Fraction First, we can use the property that a − n = a n 1 to rewrite the expression with a positive exponent: ( 81 16 ) − 4 3 = ( 16 81 ) 4 3
Expressing as a Power Next, we recognize that both 81 and 16 are perfect fourth powers: 81 = 3 4 and 16 = 2 4 . Therefore, we can rewrite the fraction as a power: 16 81 = 2 4 3 4 = ( 2 3 ) 4
Substituting Back Now, substitute this back into the expression: ( 16 81 ) 4 3 = ( ( 2 3 ) 4 ) 4 3
Simplifying the Exponent Using the property ( a m ) n = a mn , we can simplify the exponent: ( ( 2 3 ) 4 ) 4 3 = ( 2 3 ) 4 ⋅ 4 3 = ( 2 3 ) 3
Calculating the Final Result Finally, calculate the cube of the fraction: ( 2 3 ) 3 = 2 3 3 3 = 8 27 So, the simplified expression is 8 27 .
Examples
Imagine you are calculating the scaling factor for a map. If the actual area is represented by 81 16 of the map's area, and you need to find the linear scaling factor (which involves taking a fractional power), this problem demonstrates how to simplify such calculations. Understanding exponents and fractions is crucial in fields like cartography, engineering, and computer graphics, where scaling and transformations are common.
