\left(\frac{16}{81}\right)^{-\frac{3}{4}}
Answer (1)
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 expression using exponent rules: ( ( 2 3 ) 4 ) 4 3 = ( 2 3 ) 3 .
Evaluate the expression: ( 2 3 ) 3 = 8 27 .
The final answer is 8 27 .
Explanation
Understanding the Problem We are given the expression ( 81 16 ) − 4 3 . Our goal is to simplify and evaluate this expression.
Using the Negative Exponent Property First, we use the property 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 express the fraction 16 81 as a power of a fraction. We recognize that 81 = 3 4 and 16 = 2 4 , so we can write: 16 81 = 2 4 3 4 = ( 2 3 ) 4
Substitution Now, we substitute this back into our expression: ( 16 81 ) 4 3 = ( ( 2 3 ) 4 ) 4 3
Simplifying the Exponent Using the property ( a m ) n = a m ⋅ n , we simplify the exponent: ( ( 2 3 ) 4 ) 4 3 = ( 2 3 ) 4 ⋅ 4 3 = ( 2 3 ) 3
Evaluating the Expression Finally, we evaluate the expression by cubing both the numerator and the denominator: ( 2 3 ) 3 = 2 3 3 3 = 8 27 Thus, the simplified expression is 8 27 .
Final Answer The final answer is 8 27 , which can also be written as 3.375 .
Examples
Fractional exponents are used in various fields, such as calculating growth rates, determining the dimensions of objects, and modeling physical phenomena. For example, if you want to calculate the side length of a cube given its volume, you would use a fractional exponent. Similarly, in finance, fractional exponents are used to calculate compound interest rates over different time periods. Understanding fractional exponents helps in making accurate predictions and informed decisions in these real-world scenarios.
