Question

How can I write this to simple form?
`(a + 1)⁵ (a - 1)⁴ / ( a⁴ - 1) ²`
`(a + b)^n + 2 / a^ - 3 b² × a b^ - 1 c³ / (a + b)^n - 2`​

Answer 1

To simplify the given algebraic expressions, use the binomial theorem for expansion, factor the difference of squares, and cancel out common terms in the numerator and denominator. Apply rules of exponentiation for further simplification.

Step-by-step explanation:

To simplify (a + 1)⁵ (a - 1)⁴ / (a⁴ - 1)²:

1. Observe that (a - 1)⁴ can be expanded as a⁴ - 4a³ + 6a² - 4a + 1 using the binomial theorem.
2. Also, (a⁴ - 1) can be factored as (a² + 1)(a + 1)(a - 1) because it's a difference of squares.
3. Therefore, (a⁴ - 1)² is (a² + 1)²(a + 1)²(a - 1)².
4. Now you can cancel out matching terms from the numerator and the denominator to reach simpler forms.

For the expression (a + b)ⁿ + 2 / a⁻³ b² × a b⁻¹ c³ / (a + b)ⁿ - 2, apply similar expansion and factorization techniques, and then simplify by cancelling terms.

Remember that (x⁹)⁺ = x² as a rule for exponentiation to help with simplification.