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 (1)
Rewrite the cube root as a fractional exponent: ( 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 .
Compare the simplified expression with the given options.
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'll use exponent rules.
Rewriting the Cube Root First, recall that a cube root can be written as a fractional exponent: 3 a = a 3 1 . Therefore, we can rewrite the given expression as: ( 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 to our expression, we get: ( 12 5 3 1 ) x = 12 5 3 1 ⋅ x = 12 5 3 1 x
Finding the Equivalent Expression Now, we compare our simplified expression, 12 5 3 1 x , with the given options. We see that it matches the first option, 12 5 3 1 x .
Conclusion Therefore, the expression equivalent to ( 3 125 ) x is 12 5 3 1 x .
Examples
Imagine you are calculating the growth of a population over time, where the growth rate involves a cube root and an exponent. The expression ( 3 125 ) x could represent a simplified model for this growth. Understanding how to manipulate exponents allows you to predict future population sizes more easily. For example, if x represents the number of years, you can calculate the population size after a certain number of years by substituting the value of x into the equivalent expression 12 5 3 1 x . This is also applicable in financial calculations, such as compound interest, where exponents play a crucial role in determining the final amount.
