Which is equivalent to [tex]$80^{4^x}$[/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)
We analyze the given expression 8 0 4 x .
We rewrite each of the options using exponent rules.
We compare each rewritten option with the given expression.
We conclude that none of the options are equivalent to 8 0 4 x .
Therefore, there might be a typo in the question or the options.
Explanation
Understanding the Problem We are given the expression 8 0 4 x and asked to find an equivalent expression from the given choices.
Listing the Options The options are:
( 4 80 ) x
4 80 x
x 8 0 4
( x 80 ) 4
Plan of Action We need to determine which of the given options is equivalent to 8 0 4 x . Let's analyze each option using exponent rules.
Analyzing Option 1 Option 1: ( 4 80 ) x = 2 0 x . This is not equivalent to 8 0 4 x .
Analyzing Option 2 Option 2: 4 80 x = ( 8 0 4 1 ) x = 8 0 4 x . This is not equivalent to 8 0 4 x .
Analyzing Option 3 Option 3: x 8 0 4 = ( 8 0 4 ) x 1 = 8 0 x 4 . This is not equivalent to 8 0 4 x .
Analyzing Option 4 Option 4: ( x 80 ) 4 = x 4 8 0 4 . This is not equivalent to 8 0 4 x .
Conclusion After analyzing all the options, we can conclude that none of them are equivalent to the given expression 8 0 4 x . There might be a typo in the question or the options.
Examples
Understanding exponential expressions is crucial in various fields like finance, where compound interest is calculated, and in physics, where exponential decay models radioactive substances. For instance, if you invest money with continuously compounded interest, the formula A = P e r t describes how your investment grows over time, where A is the final amount, P is the principal, r is the interest rate, and t is the time. Similarly, in radioactive decay, the amount of a substance remaining after time t is given by N ( t ) = N 0 e − λ t , where N 0 is the initial amount and λ is the decay constant. These examples highlight the importance of understanding and manipulating exponential expressions.
