Question 14

The cube root of y³ equals 3⁹.

Given that the cube root of y is 3, we can express y as 3³. Therefore, when we calculate the cube root of y³, we are actually finding the cube root of (3³)³, which simplifies to 3⁹.

A) 3²

This option suggests that the cube root of y³ equals 3 squared. However, since y is equal to 3³, the cube root of y³ actually simplifies to 3⁹, making this option incorrect.

B) 3³

This option indicates that the cube root of y³ is equal to 3 cubed. Yet, based on our initial equation, the value simplifies to 3⁹, which is significantly greater than 3³. Therefore, this option is also incorrect.

C) 3⁶

This choice proposes that the cube root of y³ equals 3 to the power of 6. However, following the calculation from y = 3³, we find that the cube root of y³ is 3⁹, thus making this option incorrect as well.

D) 3⁹

This option is correct. Since y is defined as 3³, when we cube y to find y³, we get (3³)³ = 3⁹. Thus, the cube root of y³ is indeed 3⁹.

E) 3¹⁸

This option posits that the cube root of y³ equals 3 to the power of 18. This is incorrect because the value calculated from y = 3³ leads to 3⁹, which is much less than 3¹⁸.

Conclusion

The correct answer is 3⁹, as derived from the initial relationship of y being 3³. All other options misrepresent the mathematical operations applied to y, demonstrating an incorrect understanding of exponentiation and cube roots. Thus, D is the only correct solution.