Which is equivalent to [tex]$\sqrt[3]{8}^{\frac{1}{4} x}$[/tex] ?
A. [tex]$8^{\frac{3}{4} x}$[/tex]
B. [tex]$\sqrt[7]{8}^x$[/tex]
C. [tex]$\sqrt[12]{8}^x$[/tex]
D. [tex]$8^{\frac{3}{4 x}}$[/tex]
Answer (2)
Rewrite the given expression using exponent rules: 3 8 4 1 x = ( 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 as a root: 8 12 x = ( 8 12 1 ) x = 12 8 x .
The equivalent expression 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.
Rewriting the Expression We can rewrite the given expression using exponent rules. Recall that n a = a n 1 . Therefore, 3 8 = 8 3 1 . So, we have ( 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 .
Simplifying the Exponent We can rewrite the exponent 12 1 x as 12 x . So we have 8 12 x .
Rewriting as a Root We can rewrite 8 12 x as ( 8 12 1 ) x . Since 8 12 1 = 12 8 , we have ( 8 12 1 ) x = 12 8 x .
Finding the Equivalent Expression Comparing the simplified expression with the given options, we see that 12 8 x is one of the options. Therefore, the expression equivalent to 3 8 4 1 x is 12 8 x .
Examples
Understanding exponential expressions and their equivalent forms is crucial in various fields, such as calculating compound interest or modeling population growth. For instance, if a population grows at a rate proportional to its size, the population size at time t can be modeled by an exponential function. Simplifying such expressions helps in predicting future population sizes or determining growth rates.
The expression 3 8 4 1 x is equivalent to 12 8 x after applying exponent rules. Therefore, the correct answer is option C: 12 8 x .
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