\(\left(\frac{16}{81}\right)^{3 / 4}\)
Answer (1)
Rewrite the base as a ratio of powers: 81 16 = 3 4 2 4 .
Express the fraction as a single power: 3 4 2 4 = ( 3 2 ) 4 .
Apply the power of a power rule: ( ( 3 2 ) 4 ) 3/4 = ( 3 2 ) 4 ⋅ ( 3/4 ) .
Simplify and evaluate: ( 3 2 ) 3 = 27 8 . The final answer is 27 8 .
Explanation
Understanding the Problem We are asked to evaluate the expression ( 81 16 ) 3/4 . This involves understanding rational exponents and how they relate to powers and roots.
Rewriting the Base We can rewrite the base of the expression, 81 16 , as a ratio of powers. Notice that 16 = 2 4 and 81 = 3 4 . Therefore, we can rewrite the expression as ( 3 4 2 4 ) 3/4 .
Simplifying the Fraction Using the properties of exponents, we can rewrite the fraction inside the parentheses as a single power: 3 4 2 4 = ( 3 2 ) 4 . So, our expression becomes ( ( 3 2 ) 4 ) 3/4 .
Applying the Power of a Power Rule Now, we apply the power of a power rule, which states that ( a m ) n = a m ⋅ n . In our case, this means ( ( 3 2 ) 4 ) 3/4 = ( 3 2 ) 4 ⋅ ( 3/4 ) .
Simplifying the Exponent Next, we simplify the exponent: 4 ⋅ ( 3/4 ) = 3 . Thus, our expression simplifies to ( 3 2 ) 3 .
Evaluating the Final Expression Finally, we evaluate the expression by raising both the numerator and the denominator to the power of 3: ( 3 2 ) 3 = 3 3 2 3 = 27 8 .
Final Answer Therefore, ( 81 16 ) 3/4 = 27 8 .
Examples
Understanding rational exponents is crucial in various fields, such as physics and engineering, where quantities often scale with fractional powers. For instance, the period of a pendulum scales with the square root of its length, and understanding such relationships helps engineers design accurate timekeeping devices. Similarly, in fluid dynamics, drag forces can depend on fractional powers of velocity, which is essential for designing efficient vehicles and aircraft. This problem demonstrates how to simplify and evaluate expressions with rational exponents, a skill that is invaluable in these practical applications.
