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 (1)
Rewrite the given expression using fractional exponents: 4 9 2 1 x = ( 9 4 1 ) 2 1 x .
Apply the power of a power rule: ( a m ) n = a m ⋅ n to simplify the expression: ( 9 4 1 ) 2 1 x = 9 4 1 ⋅ 2 1 x = 9 8 1 x .
Compare the simplified expression with the given options.
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 We can rewrite the given expression using fractional exponents. Recall that n a = a n 1 . Thus, 4 9 = 9 4 1 . So, the given expression becomes ( 9 4 1 ) 2 1 x .
Applying the Power of a Power Rule Now, we use the power of a power rule, which states that ( a m ) n = a m ⋅ n . Applying this rule, we have ( 9 4 1 ) 2 1 x = 9 4 1 ⋅ 2 1 x = 9 8 1 x .
Identifying the Equivalent Expression Comparing the simplified expression 9 8 1 x with the given options, we see that it matches the second option, 9 8 1 x . Therefore, the equivalent expression is 9 8 1 x .
Examples
Understanding exponential expressions and their manipulations is crucial in various fields, such as finance and computer science. For instance, calculating compound interest involves exponential growth, where the initial investment grows over time. Similarly, in computer science, analyzing the time complexity of algorithms often involves exponential functions. For example, if an algorithm's time complexity is O ( 2 n ) , understanding how to simplify and manipulate exponential expressions helps in assessing the algorithm's efficiency and scalability. The ability to simplify expressions like 4 9 2 1 x to 9 8 1 x can be directly applied in these scenarios to make calculations and comparisons easier.
