Simplify $(6^{1 / 9})^4$.
A. $6^{16}$
B. 1
C. $6^4$
D. 6
Answer (2)
Apply the power of a power rule: ( a m ) n = a m × n .
Simplify the expression: ( 6 1/9 ) 4 = 6 ( 1/9 ) × 4 = 6 4/9 .
Compare the simplified expression with the given options.
The correct simplification is 6 4/9 , which is not among the options. There is no correct answer among the options.
If forced to choose the closest option, it would be 6, but this is just an approximation.
The final answer is that there is no correct answer among the options. However, the simplified expression is 6 4/9 .
Explanation
Understanding the problem We are given the expression ( 6 1/9 ) 4 and asked to simplify it. We need to use the power of a power rule, which states that ( a m ) n = a m × n .
Applying the power of a power rule Applying the power of a power rule to the given expression, we get: ( 6 1/9 ) 4 = 6 ( 1/9 ) × 4 = 6 4/9 So the simplified expression is 6 4/9 .
Comparing with the options Now we need to compare our simplified expression 6 4/9 with the given options: A. 6 16 B. 1 C. 6 4 D. 6 We can see that none of the options match our simplified expression 6 4/9 . However, we can approximate the value of 6 4/9 to see if it is close to any of the options. Using a calculator, we find that 6 4/9 ≈ 2.217 .
Final Answer Since none of the options match the simplified expression, it is possible that there was a typo in the question or the options. However, based on the given options, we can conclude that the correct simplification of the given expression is 6 4/9 , which is not among the options.
Additional Consideration Based on the options provided, none of them are equivalent to 6 4/9 . However, if we were forced to choose the closest option, we could consider that 6 4/9 ≈ 2.217 , which is closest to the value of 6. But this is just an approximation and not an exact match.
Examples
Understanding exponential expressions like this is useful in many areas, such as calculating compound interest, where the exponent represents the number of compounding periods. For example, if you invest money at a certain interest rate compounded quarterly, you would use exponents to calculate the future value of your investment. Similarly, in science, exponential expressions are used to model population growth or radioactive decay.
The expression ( 6 1/9 ) 4 simplifies to 6 4/9 , which is not among the provided options. Thus, the correct simplification is 6 4/9 . Therefore, there is no correct answer among the options given.
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