Question

Factor the fourth root of 162xy⁵z⁻⁸. a) 3xy⁵/⁴ z⁻² b) 3x⁴y⁵/⁴ z⁻² c) 3xy⁵/¹⁶ z⁻² d) 3x⁴y⁵/¹⁶ z⁻²

Answer 1

To factor the fourth root of 162xy^5z^-8, break down 162 into prime factors and apply the fourth root to each part. The expression simplifies to 3xy^5/4z^-2, corresponding to option (a).

Step-by-step explanation:

To factor the fourth root of 162xy5z−8, we must first simplify the expression under the radical by breaking down 162 into its prime factors and then applying the fourth root to each part.

162 can be factored into 2 × 34. So, we have:

∛(2 × 34 × x × y5 × z−8)

Since we're taking the fourth root, we look for groups of four:

• 34 becomes 3 since the fourth root of 34 is 3.
• y5 can be split into y4 × y, and the fourth root of y4 is y.
• z−8 as z−2 × z−6, where the fourth root of z−2 is z−2/4 or z−2.

However, we cannot take the fourth root of 2 without it remaining under the radical. Therefore, the factored form of the expression is:

3xy5/4z−2

This corresponds to option (a). All parts of the expression except for the 2 are raised to an exponent that is a multiple of 4, allowing us to take them out of the fourth root.