Which is equivalent to $\sqrt[4]{9}^{\frac{1}{2} x}$ ?
A. $9^{2 x}$
B. $9^{\frac{1}{8} x}$
C. $\sqrt{9}^x$
D. $\sqrt[6]{9}^x$
Answer (2)
Rewrite the fourth root of 9 as a fractional exponent: 4 9 = 9 4 1 .
Substitute this into the original expression: 4 9 2 1 x = ( 9 4 1 ) 2 1 x .
Apply the power of a power rule: ( a m ) n = a mn , so ( 9 4 1 ) 2 1 x = 9 4 1 ⋅ 2 1 x = 9 8 1 x .
The equivalent expression is 9 8 1 x .
Explanation
Understanding the Problem We are given the expression 4 9 2 1 x and asked to find an equivalent expression from the options: 9 2 x , 9 8 1 x , 9 x , 6 9 x .
Rewriting with Fractional Exponents First, let's rewrite the given expression using fractional exponents. Recall that n a = a n 1 . Therefore, 4 9 = 9 4 1 .
Substituting Back Now, substitute this back into the original expression: 4 9 2 1 x = ( 9 4 1 ) 2 1 x .
Applying the Power of a Power Rule Using the power of a power rule, which states that ( a m ) n = a mn , we can simplify the expression: ( 9 4 1 ) 2 1 x = 9 4 1 ⋅ 2 1 x = 9 8 1 x .
Identifying the Equivalent Expression Finally, we compare the simplified expression 9 8 1 x with the given options. We see that it matches the second option, 9 8 1 x .
Examples
Understanding exponential expressions and their manipulations is crucial in various fields such as finance, physics, 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 understanding such expressions is essential for financial planning and analysis.
The expression 4 9 2 1 x simplifies to 9 8 1 x , which is equivalent to option B. This is achieved by rewriting the fourth root of 9 as a fractional exponent, applying the power of a power rule, and then identifying the equivalent expression among the choices given.
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