Question

Сократите дроби m^2-6m+9/m^3-9m a^2+ab+b^2/a^6-a^3b^3

Answer 1

Решение внизу на фото

Answer 2

To simplify the given expression, let's break it down into separate fractions and then simplify each one individually:

1. Simplify (m^2 - 6m + 9) / (m^3 - 9m): Factor the numerator and denominator:

Numerator: m^2 - 6m + 9 = (m - 3)^2 Denominator: m^3 - 9m = m(m^2 - 9) = m(m - 3)(m + 3)

Cancel out common factors:

(m^2 - 6m + 9) / (m^3 - 9m) = (m - 3)^2 / [m(m - 3)(m + 3)]

2. Simplify (a^2 + ab + b^2) / (a^6 - a^3b^3): Factor the denominator:

Denominator: a^6 - a^3b^3 = a^3(a^3 - b^3)

Now, recall the identity: a^3 - b^3 = (a - b)(a^2 + ab + b^2)

So, the denominator becomes: a^6 - a^3b^3 = a^3(a - b)(a^2 + ab + b^2)

Cancel out common factors:

(a^2 + ab + b^2) / (a^6 - a^3b^3) = 1 / [a^3(a - b)]

Putting it all together:

The simplified expression is [(m - 3)^2 / (m(m - 3)(m + 3))] * [1 / (a^3(a - b))].

Now, look for common factors to cancel further, but based on the provided expression, this is the simplest form.