Which is equivalent to $\sqrt[3]{8}^{\frac{1}{4} x}$ ?
A. $8^{\frac{3}{4} x}$
B. $\sqrt[7]{8}^x$
C. $\sqrt[12]{8^x}$
D. $8^{\frac{3}{4 x}}$
Answer (1)
Simplify the base: 3 8 = 2 , so the expression becomes 2 4 1 x .
Change the base to 8: 2 = 8 3 1 , so the expression becomes ( 8 3 1 ) 4 1 x .
Apply the power of a power rule: ( 8 3 1 ) 4 1 x = 8 12 1 x .
Rewrite the expression: 8 12 1 x = 12 8 x . The final answer is 12 8 x .
Explanation
Understanding the Problem We are given the expression 3 8 4 1 x and asked to find an equivalent expression from the given options: 8 4 3 x , 7 8 x , 12 8 x , 8 4 x 3 .
Simplifying the Base First, we simplify the base of the expression. We know that 3 8 = 2 , since 2 3 = 8 . So, we can rewrite the given expression as 2 4 1 x .
Changing the Base Now, we want to express this in terms of base 8. Since 2 = 8 3 1 , we can substitute this into our expression: 2 4 1 x = ( 8 3 1 ) 4 1 x .
Applying the Power of a Power Rule Using the power of a power rule, ( a m ) n = a mn , we have ( 8 3 1 ) 4 1 x = 8 3 1 ⋅ 4 1 x = 8 12 1 x .
Rewriting the Expression We can rewrite 8 12 1 x as ( 8 x ) 12 1 , which is equivalent to 12 8 x .
Finding the Equivalent Expression Comparing our simplified expression 12 8 x with the given options, we see that it matches the third option.
Final Answer Therefore, the expression equivalent to 3 8 4 1 x is 12 8 x .
Examples
Understanding exponential expressions and their manipulations is crucial in various fields, such as calculating compound interest, modeling population growth, and analyzing radioactive decay. For instance, if you invest money in an account with continuously compounded interest, the formula for the amount you'll have after a certain time involves exponential expressions. Similarly, in radioactive decay, the amount of a substance remaining after a certain time is modeled by an exponential function. By mastering these concepts, you can make informed financial decisions and understand natural phenomena.
