Need help. Thank you!

Question

1. Find the error in the simplified radical process shown below. Explain the error.

2. Simplify the expression from question 1 correctly.

3. Solve the equation x⁴ + 81 = 3831. Find the exact answer in simplified radical form.

Answer 1

Take a close look at the first step. Remember that taking the cube root of something is the same as raising something to the 1/3 power, so that you could re-write the whole thing as:

(2⁴x³/y)^1/3

Further remember that when you raise an exponent to a power, you multiply the exponents together. In general, this means (x^a)^b would equal x^ab. To use some actual numbers, (2³)⁴ = 2¹².

Pulling the x out of the cube root is perfectly fine, because (x³)^1/3 = x^{3 * 1/3} = x¹ = x. But how would you want to simplify (2⁴)^1/3? That's where you need to look to find the answer to parts 1 and 2.