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 (1)
Rewrite the expression using the exponent rule ( a b ) c = a b c in reverse: 8 0 4 1 x = ( 8 0 4 1 ) x .
Recognize that 8 0 4 1 is the same as 4 80 .
Therefore, 8 0 4 1 x = ( 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 provided options.
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 . This is because 4 1 x = 4 1 ⋅ x .
Using Radical Notation We recognize that 8 0 4 1 is the same as 4 80 . Therefore, we have 8 0 4 1 x = ( 8 0 4 1 ) x = ( 4 80 ) x .
Finding the Equivalent Expression Comparing this result with the given options, we see that 4 80 x is one of the options.
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, when calculating compound interest, the formula often involves raising a base (1 + interest rate) to the power of time. Similarly, in physics, exponential decay models, like those used for radioactive substances, rely on understanding how quantities change with fractional exponents. This problem reinforces the ability to manipulate exponents, which is fundamental to solving real-world problems involving growth and decay.
