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 (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: 8 12 1 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 options: 8 4 3 x , 7 8 x , 12 8 x , 8 4 x 3 .
Rewriting the Expression We can rewrite the given expression using exponent rules. Recall that n a = a n 1 . Therefore, 3 8 4 1 x = ( 8 3 1 ) 4 1 x .
Applying the Power of a Power Rule Using the power of a power rule, which states that ( a m ) n = a mn , we can simplify the expression further: ( 8 3 1 ) 4 1 x = 8 3 1 ⋅ 4 1 x = 8 12 1 x .
Rewriting with a Root Now, we can rewrite 8 12 1 x as ( 8 12 1 ) x . Since 8 12 1 is the same as 12 8 , we have ( 8 12 1 ) x = 12 8 x .
Finding the Equivalent Expression Comparing the simplified expression 12 8 x with the given options, we see that it matches the option 12 8 x . Therefore, the equivalent expression is 12 8 x .
Examples
Imagine you're adjusting the volume on a speaker. The original volume is represented by 3 8 4 1 x . By simplifying this expression to 12 8 x , you're essentially finding a more straightforward way to describe the same volume level. This is similar to how sound engineers might use different but equivalent formulas to fine-tune audio settings for optimal performance. Understanding exponent rules allows for easier manipulation and comprehension of these settings.
The expression 3 8 4 1 x simplifies to 12 8 x . This follows from using exponent rules to rewrite the expression correctly. Therefore, the equivalent expression is 12 8 x .
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