Which is equivalent to [tex]$\sqrt[5]{1,215}^x$[/tex]?
A. [tex]$243^x$[/tex]
B. [tex]$1,215^{\frac{1}{5} x}$[/tex]
C. [tex]$1,215^{\frac{1}{5 x}}$[/tex]
D. [tex]$243^{\frac{1}{x}}$[/tex]
Answer (1)
Rewrite the given expression using fractional exponents: 5 1 , 215 x = ( 1 , 21 5 5 1 ) x .
Apply the power of a power rule: ( 1 , 21 5 5 1 ) x = 1 , 21 5 5 1 x .
Simplify the exponent: 1 , 21 5 5 1 x .
The equivalent expression is 1 , 21 5 5 1 x .
Explanation
Understanding the Problem We are given the expression 5 1 , 215 x and asked to find an equivalent expression from the given choices: 24 3 x , 1 , 21 5 5 1 x , 1 , 21 5 5 x 1 , 24 3 x 1 .
Rewriting the Expression We can rewrite the given expression using the properties of exponents. Recall that n a = a n 1 . Therefore, 5 1 , 215 = 1 , 21 5 5 1 .
Applying the Power of a Power Rule Now we have 5 1 , 215 x = ( 1 , 21 5 5 1 ) x . Using the power of a power rule, which states that ( a m ) n = a m × n , we get ( 1 , 21 5 5 1 ) x = 1 , 21 5 5 1 × x = 1 , 21 5 5 x .
Simplifying the Exponent We can also write 5 x as 5 1 x , so we have 1 , 21 5 5 x = 1 , 21 5 5 1 x .
Finding the Equivalent Expression Comparing this with the given choices, we see that 1 , 21 5 5 1 x matches the second choice: 1 , 21 5 5 1 x . Therefore, the equivalent expression is 1 , 21 5 5 1 x .
Examples
Understanding and manipulating exponents is crucial in many fields, such as finance when calculating compound interest. For example, if you invest P dollars at an annual interest rate r compounded n times per year, the amount A you'll have after t years is given by A = P ( 1 + n r ) n t . This formula uses exponents to calculate the growth of your investment over time. Similarly, in physics, exponential functions are used to model radioactive decay, where 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.
