Simplify the rational expression. state any restrictions on the variable. 
n⁴ - 10n² + 24/n⁴ - 9n² + 18

Solution:

Given rational expression is n⁴ - 10n² + 24/n⁴ - 9n² + 18

Rewrite n⁴ as (n²)²

(n²)² - 10n² + 24/ n⁴ - 9n² + 18

Let u = n². Substitute u for all occurrences of n²

(u² - 10u + 24) / n⁴ - 9n² + 18

Consider the terms within the brackets it is in the form x² + bx + c.

Find a pair of integers whose product is c and whose sum is b.

 In this case, whose product is 24 and whose sum is -10

The factors are -6, -4

Write the factored form using these integers.

(u - 6)(u - 4) / n⁴ - 9n² + 18

(n² - 6)(n² - 22) / n⁴ - 9n² + 18

(n² - 6)(n + 2)(n - 2) / n⁴ - 9n² + 18

(n² - 6)(n + 2)(n - 2) / (n² - 6)(n² - 3)

(n + 2)(n - 2)/n² - 3

This expression is valid only when n² - 3 ≠ 0

So, the restriction to the variable n is, it shouldn’t be equal to √3.

Therefore, the simplified expression is (n + 2)(n - 2)/n² - 3 and the restriction is n ≠ √3

---

Simplify the rational expression. state any restrictions on the variable.
n⁴ - 10n² + 24/n⁴ - 9n² + 18

Summary:

The simplified expression of n⁴ - 10n² + 24/n⁴ - 9n² + 18 is (n + 2)(n - 2)/n² - 3 and the restriction is n ≠ √3