Simplify the expression.
$512^{-\frac{5}{3}}$
Simplify $512^{-\frac{5}{3}}$ completely. Choose the correct answer below.
A. $\sqrt[3]{512^{-5}}$
B. -32768
C. $\frac{1}{(\sqrt[3]{512})^5}$
D. $\frac{1}{32768}$
Answer (1)
Rewrite the expression using the property of negative exponents: 51 2 − 3 5 = 51 2 3 5 1 .
Rewrite the expression using the property of fractional exponents: 51 2 3 5 1 = ( 3 512 ) 5 1 .
Calculate the cube root of 512: 3 512 = 8 .
Calculate 8 5 and simplify: 8 5 1 = 32768 1 . The final answer is 32768 1 .
Explanation
Understanding the Problem We are given the expression 51 2 − 3 5 and asked to simplify it. We need to choose the correct answer from the given options.
Using Negative Exponent Property First, let's rewrite the expression using the property a − n = a n 1 . This gives us: 51 2 − 3 5 = 51 2 3 5 1
Using Fractional Exponent Property Next, we use the property a n m = ( n a ) m . Applying this to our expression, we get: 51 2 3 5 1 = ( 3 512 ) 5 1
Calculating the Cube Root Now, we need to find the cube root of 512. We know that 8 3 = 512 , so 3 512 = 8 . Substituting this value into our expression, we have: ( 3 512 ) 5 1 = 8 5 1
Calculating the Power Now, we calculate 8 5 . 8 5 = 8 × 8 × 8 × 8 × 8 = 32768 So our expression becomes: 8 5 1 = 32768 1
Final Answer Therefore, the simplified expression is 32768 1 . Comparing this with the given options, we see that option D is the correct answer.
Examples
Understanding exponents and roots is crucial in many scientific and engineering fields. For example, when calculating the decay rate of radioactive materials, you often encounter exponential functions with fractional exponents. Simplifying these expressions allows scientists to accurately predict the remaining amount of a substance over time, which is vital in nuclear medicine and environmental science.
