Evaluate: $\left(\frac{64}{27}\right)^{\frac{2}{3}}$
Answer (2)
The expression ( 27 64 ) 3 2 evaluates to 9 16 by first taking the cube root and then squaring the result. The cube root of 27 64 is 3 4 , which when squared gives 9 16 .
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Rewrite the expression using exponent rules: ( 27 64 ) 3 2 = (( 27 64 ) 3 1 ) 2 .
Evaluate the cube root: ( 27 64 ) 3 1 = 3 4 .
Square the result: ( 3 4 ) 2 = 9 16 .
The final answer is 9 16 .
Explanation
Understanding the Problem We are asked to evaluate the expression ( 27 64 ) 3 2 . This involves understanding fractional exponents and how they relate to roots and powers.
Rewriting the Expression We can rewrite the expression using the properties of exponents. Specifically, we can express the fractional exponent as a combination of a root and a power: ( 27 64 ) 3 2 = ( ( 27 64 ) 3 1 ) 2 This means we first take the cube root of 27 64 , and then square the result.
Evaluating the Cube Root Now, let's find the cube root of 27 64 . We need to find a number that, when multiplied by itself three times, equals 27 64 . Since 4 3 = 64 and 3 3 = 27 , we have ( 27 64 ) 3 1 = 3 4
Squaring the Result Next, we square the result from the previous step: ( 3 4 ) 2 = 3 2 4 2 = 9 16
Final Answer Therefore, the value of the expression ( 27 64 ) 3 2 is 9 16 .
Examples
Fractional exponents are useful in various fields, such as physics and engineering, when dealing with quantities that change according to power laws. For example, the period of a simple pendulum is proportional to the square root of its length, which can be expressed using a fractional exponent. Similarly, in finance, compound interest calculations often involve fractional exponents to determine the growth of investments over time. Understanding fractional exponents allows us to model and analyze these real-world phenomena effectively.
