Which is equivalent to $\sqrt{10}^{\frac{3}{4} x}$ ?
A. $(\sqrt[3]{10})^{4 x}$
B. $(\sqrt[4]{10})^{3 x}$
C. $(\sqrt[6]{10})^{4 x}$
D. $(\sqrt[8]{10})^{3 x}$
Answer (1)
Rewrite the given expression 10 4 3 x as 1 0 8 3 x .
Rewrite each of the options in the form 1 0 k for some k .
Compare the exponent of 10 in the rewritten given expression with the exponent of 10 in each of the rewritten options.
The equivalent expression is ( 8 10 ) 3 x , since it can be rewritten as 1 0 8 3 x .
The final answer is ( 8 10 ) 3 x
Explanation
Understanding the Problem Let's analyze the problem. We are given the expression 10 4 3 x and we need to find an equivalent expression from the given options.
Rewriting the Given Expression First, let's rewrite the given expression using exponent rules. Recall that a = a 2 1 . Therefore, we can rewrite the given expression as follows: 10 4 3 x = ( 1 0 2 1 ) 4 3 x Now, using the rule ( a m ) n = a m ⋅ n , we have: ( 1 0 2 1 ) 4 3 x = 1 0 2 1 ⋅ 4 3 x = 1 0 8 3 x
Rewriting the Options Now, let's rewrite each of the options in the form 1 0 k for some k and compare them with the rewritten given expression.
Option 1: ( 3 10 ) 4 x
Recall that n a = a n 1 . So, 3 10 = 1 0 3 1 . Therefore, ( 3 10 ) 4 x = ( 1 0 3 1 ) 4 x = 1 0 3 1 ⋅ 4 x = 1 0 3 4 x
Option 2: ( 4 10 ) 3 x
Similarly, 4 10 = 1 0 4 1 . Therefore, ( 4 10 ) 3 x = ( 1 0 4 1 ) 3 x = 1 0 4 1 ⋅ 3 x = 1 0 4 3 x
Option 3: ( 6 10 ) 4 x
We have 6 10 = 1 0 6 1 . Therefore, ( 6 10 ) 4 x = ( 1 0 6 1 ) 4 x = 1 0 6 1 ⋅ 4 x = 1 0 6 4 x = 1 0 3 2 x
Option 4: ( 8 10 ) 3 x
We have 8 10 = 1 0 8 1 . Therefore, ( 8 10 ) 3 x = ( 1 0 8 1 ) 3 x = 1 0 8 1 ⋅ 3 x = 1 0 8 3 x
Finding the Equivalent Expression Comparing the exponent of 10 in the rewritten given expression with the exponent of 10 in each of the rewritten options, we see that the exponent of 10 in the given expression is 8 3 x . Option 4 has the same exponent, which is 8 3 x . Therefore, the equivalent expression is ( 8 10 ) 3 x .
Final Answer Therefore, the expression equivalent to 10 4 3 x is ( 8 10 ) 3 x .
Examples
Understanding exponential expressions is crucial in various fields, such as calculating compound interest. For instance, if you invest money in an account with continuously compounded interest, the formula for the future value of your investment involves exponential expressions. Suppose you invest P dollars at an annual interest rate r compounded continuously for t years. The future value A of your investment is given by A = P e r t , where e is the base of the natural logarithm. Simplifying and manipulating such exponential expressions helps in financial planning and understanding investment growth.
