Problem: Find x^4 and 1/x^4 given x=root3 root2/root3-root2.

Solution:

Step 1: Simplify x by rationalizing the denominator.

We have:

x = root3 root2 / (root3 - root2)

Multiplying the numerator and denominator by (root3 + root2), we get:

x = (root3 root2) (root3 + root2) / [(root3 - root2) (root3 + root2)]

x = (root6 + root6) / (root9 - root4)

x = 2 root6 / root5

Step 2: Find x^4

We have:

x^4 = (2 root6 / root5)^4

x^4 = 16 root6^4 / root5^4

x^4 = 16 (root6^2)^2 / (root5^2)^2

x^4 = 16 (6) / 25

x^4 = 96 / 25

Step 3: Find 1/x^4

We have:

1/x^4 = 1 / (96/25)

1/x^4 = 25 / 96

Therefore, x^4 = 96/25 and 1/x^4 = 25/96.

Explanation:

The problem requires us to find x^4 and 1/x^4 given x. We first simplify x by rationalizing the denominator. We then use the simplified expression to find x^4 and 1/x^4. The solution involves basic algebraic manipulation and simplification of radicals.

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