Which is equivalent to $\sqrt[5]{1,215}^x$ ?
A. $243^x$
B. $1,215^{\frac{1}{5} x}$
C. $1,215^{\frac{1}{5 x}}$
D. $243^{\frac{1}{x}}$
Answer (2)
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 x .
Simplify the exponent: 1 , 21 5 5 x = 1 , 21 5 5 1 x .
Identify the matching expression from the choices: 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 1 , 21 5 5 x as 1 , 21 5 5 1 x . Now we compare this result with the given choices.
Identifying the Matching Expression The choices are: 24 3 x 1 , 21 5 5 1 x 1 , 21 5 5 x 1 24 3 x 1 We see that 1 , 21 5 5 1 x matches our simplified expression.
Final Answer Therefore, the expression equivalent to 5 1 , 215 x is 1 , 21 5 5 1 x .
Examples
Understanding and manipulating exponents is crucial in various fields, such as finance and computer science. For instance, when calculating compound interest, the formula involves raising the interest rate plus one to the power of the number of compounding periods. Similarly, in computer science, exponents are used to describe the growth rate of algorithms, where an algorithm with a time complexity of O ( n 2 ) takes time proportional to the square of the input size n . These examples demonstrate how a solid grasp of exponents can help in making informed decisions and optimizing processes in real-world scenarios.
The expression 5 1 , 215 x simplifies to 1 , 21 5 5 1 x , making option B the correct answer. Thus, the equivalent expression is 1 , 21 5 5 1 x .
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