Which is equivalent to $(\sqrt[3]{125})^x$?
A. $125^{\frac{1}{3} x}$
B. $125^{\frac{1}{3 x}}$
C. $125^{3 x}$
D. $125^{\left(\frac{1}{3}\right)^x}$
Answer (2)
Rewrite the cube root as a fractional exponent: 3 125 = 12 5 3 1 .
Substitute back into the original expression: ( 3 125 ) x = ( 12 5 3 1 ) x .
Apply the power of a power rule: ( 12 5 3 1 ) x = 12 5 3 1 x .
The equivalent expression is 12 5 3 1 x .
Explanation
Understanding the problem We are given the expression ( 3 125 ) x and asked to find an equivalent expression from the options: 12 5 3 1 x , 12 5 3 x 1 , 12 5 3 x , 12 5 ( 3 1 ) x . To solve this, we will use exponent rules.
Rewriting the cube root First, let's rewrite the cube root as a fractional exponent. Recall that n a = a n 1 . Therefore, 3 125 = 12 5 3 1 .
Substituting back Now, substitute this back into the original expression: ( 3 125 ) x = ( 12 5 3 1 ) x .
Applying the power of a power rule Next, we use the power of a power rule, which states that ( a m ) n = a mn . Applying this rule, we get ( 12 5 3 1 ) x = 12 5 3 1 x .
Finding the equivalent expression Comparing this result with the given options, we see that 12 5 3 1 x is one of the options. Therefore, the expression equivalent to ( 3 125 ) x is 12 5 3 1 x .
Final Answer The equivalent expression is 12 5 3 1 x .
Examples
Understanding exponent rules is crucial in various fields, such as calculating compound interest, where the amount of money grows exponentially over time. For example, if you invest P dollars at an annual interest rate r compounded n times per year, the amount A after t years is given by A = P ( 1 + n r ) n t . This formula uses the power of a power rule, similar to the one used in this problem, to calculate the final amount. Simplifying such expressions helps in financial planning and understanding growth patterns.
The expression equivalent to ( 3 125 ) x is 12 5 3 1 x , which matches option A. We rewrote the cube root as a fractional exponent and applied the power of a power rule to arrive at this result.
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