Which is equivalent to [tex] \sqrt{10}^{\frac{3}{4} x} [/tex]?
A. [tex] (\sqrt[3]{10})^{4 x} [/tex]
B. [tex] (\sqrt[4]{10})^{3 x} [/tex]
C. [tex] (\sqrt[6]{10})^{4 x} [/tex]
D. [tex] (\sqrt[8]{10})^{3 x} [/tex]
Answer (2)
Rewrite the given expression using fractional exponents: 10 4 3 x = 1 0 8 3 x .
Rewrite each of the options using fractional exponents.
Compare the exponent of 10 in the rewritten given expression with the exponents of 10 in the rewritten options.
The expression equivalent to 10 4 3 x is ( 8 10 ) 3 x . ( 8 10 ) 3 x
Explanation
Understanding the Problem We are given the expression 10 4 3 x and asked to find an equivalent expression among the given options.
Rewriting the Given Expression First, let's rewrite the given expression using fractional exponents. Recall that a = a 2 1 . Therefore, 10 = 1 0 2 1 . So we have: 10 4 3 x = ( 1 0 2 1 ) 4 3 x Using the property of exponents ( a m ) n = a m × n , we get: ( 1 0 2 1 ) 4 3 x = 1 0 2 1 × 4 3 x = 1 0 8 3 x So the given expression is equivalent to 1 0 8 3 x .
Rewriting the Options Now, let's rewrite each of the options using fractional exponents and the property n a = a n 1 :
Option 1: ( 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 = ( 1 0 4 1 ) 3 x = 1 0 4 1 × 3 x = 1 0 4 3 x
Option 3: ( 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 = ( 1 0 8 1 ) 3 x = 1 0 8 1 × 3 x = 1 0 8 3 x
Comparing the Exponents Comparing the exponent of 10 in the rewritten given expression with the exponents of 10 in the rewritten options, we see that the exponent of 10 in the given expression is 8 3 x , and the exponent of 10 in option 4 is also 8 3 x . Therefore, option 4 is the equivalent expression.
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 like finance and computer science. For instance, calculating compound interest involves exponential growth. If you invest an amount P at an annual interest rate r compounded n times per year, the amount A after t years is given by A = P ( 1 + n r ) n t . Simplifying and manipulating such expressions using exponent rules, similar to what we did in the problem, helps in predicting investment growth and making informed financial decisions. Also, in computer science, understanding exponential growth is important in analyzing the complexity of algorithms.
The expression equivalent to 10 4 3 x is ( 8 10 ) 3 x . This was determined by rewriting the original expression and comparing it with the options provided. Option D is the correct choice as it matches the rewritten expression's exponent.
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