Which is equivalent to [tex]$80^{\frac{1}{4} x}$[/tex]?
A. [tex]$\left(\frac{80}{4}\right)^x$[/tex]
B. [tex]$\sqrt[4]{80}^x$[/tex]
C. [tex]$\sqrt[x]{80^4}$[/tex]
D. [tex]$\left(\frac{80}{x}\right)^4$[/tex]
Answer (2)
Rewrite the expression using exponent rules: 8 0 4 1 x = ( 8 0 4 1 ) x .
Recognize that 8 0 4 1 is the same as 4 80 .
Therefore, the expression becomes ( 4 80 ) x .
The equivalent expression is 4 80 x .
Explanation
Understanding the Problem We are given the expression 8 0 4 1 x and asked to find an equivalent expression from the given choices.
Applying Exponent Rules We can rewrite the given expression using exponent rules. Specifically, we will use the rule ( a b ) c = a b c in reverse.
Rewriting the Expression We rewrite 8 0 4 1 x as ( 8 0 4 1 ) x .
Simplifying the Expression We recognize that 8 0 4 1 is the same as 4 80 . Therefore, the expression becomes ( 4 80 ) x .
Finding the Equivalent Expression Comparing this result with the given choices, we see that the equivalent expression is 4 80 x .
Final Answer Therefore, the expression equivalent to 8 0 4 1 x is 4 80 x .
Examples
Understanding exponential expressions is crucial in various fields, such as finance and physics. For instance, calculating compound interest involves exponential growth. If you invest an amount P 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 demonstrates how exponential expressions are used to model real-world phenomena.
The equivalent expression to 8 0 4 1 x is 4 80 x , which corresponds to option B. We rewrite the original expression using exponent rules to simplify it. This allows us to easily find the equivalent form among the choices given.
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