Which is equivalent to $\sqrt[4]{9}^{\frac{1}{2} x}$ ?
A. $9^{2 x}$
B. $9^{\frac{1}{6} x}$
C. $\sqrt{9}{ }^x$
D. $\sqrt[6]{9} x^x$
Answer (1)
Rewrite the original expression using exponents: 4 9 2 1 x = ( 9 4 1 ) 2 1 x .
Apply the power of a power rule: ( 9 4 1 ) 2 1 x = 9 8 1 x .
Compare the result with the given options.
The closest equivalent expression is 9 6 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 given options: 9 2 x , 9 6 1 x , 9 x , 6 9 x x .
Rewriting with Exponents First, let's rewrite the given expression using exponents. Recall that n a = a n 1 . Thus, 4 9 = 9 4 1 . Substituting this into the original expression, we get ( 9 4 1 ) 2 1 x .
Applying Power of a Power Rule Now, we use the power of a power rule, which states that ( a m ) n = a mn . Applying this rule, we have ( 9 4 1 ) 2 1 x = 9 4 1 ⋅ 2 1 x = 9 8 1 x .
Analyzing the Options Now, let's examine the given options and see if any of them are equivalent to 9 8 1 x .
Option 1: 9 2 x Option 2: 9 6 1 x Option 3: 9 x = ( 9 2 1 ) x = 9 2 1 x = 9 2 x Option 4: 6 9 x x = 9 6 1 x x
Comparing and Concluding Comparing the options to 9 8 1 x , we see that Option 2, 9 6 1 x , is the closest. Although it is not exactly the same, it is the only option that has a similar form. The other options have different exponents or an additional x x term.
Final Answer Therefore, the closest equivalent expression to 4 9 2 1 x is 9 6 1 x .
Examples
Understanding exponential expressions is crucial in various fields, such as finance and computer science. For example, when calculating compound interest, the formula involves raising a base (1 + interest rate) to the power of time. Similarly, in computer science, exponential growth is used to describe the complexity of algorithms. Knowing how to manipulate and simplify exponential expressions allows us to model and analyze these real-world phenomena effectively, making informed decisions and predictions.
