# How do you simplify #(x^3+8)*(x-2)/(x^2-2x+4)div(x^2-4)/(x-6)#?

In algebraic fractions you want to factorize as much as possible.

Now that everything is expressed as factors you may cancel:

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To simplify the expression, we can start by factoring the numerator and denominator separately.

The numerator, (x^3 + 8), can be factored using the sum of cubes formula: a^3 + b^3 = (a + b)(a^2 - ab + b^2). In this case, a = x and b = 2. So, (x^3 + 8) can be factored as (x + 2)(x^2 - 2x + 4).

The denominator, (x^2 - 2x + 4), cannot be factored further.

Next, we can factor the denominator of the division, (x^2 - 4), as the difference of squares: a^2 - b^2 = (a + b)(a - b). In this case, a = x and b = 2. So, (x^2 - 4) can be factored as (x + 2)(x - 2).

Now, we can rewrite the expression as [(x + 2)(x^2 - 2x + 4)] / [(x + 2)(x - 2)].

The (x + 2) terms in the numerator and denominator cancel out, leaving us with (x^2 - 2x + 4) / (x - 2).

Therefore, the simplified expression is (x^2 - 2x + 4) / (x - 2).

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To simplify the expression (\frac{(x^3+8)(x-2)}{x^2-2x+4} \div \frac{x^2-4}{x-6}), you can follow these steps:

- Factorize the expressions in both the numerator and denominator.
- Rewrite the division as multiplication by the reciprocal of the second fraction.
- Simplify by canceling out common factors.

Starting with the numerator and denominator factorizations:

((x^3+8) = (x+2)(x^2-2x+4))

((x^2-4) = (x+2)(x-2))

((x^2-2x+4)) remains as it is.

Now, rewrite the expression with factored forms:

(\frac{(x+2)(x-2)(x-2)}{(x+2)(x^2-2x+4)} \times \frac{(x-6)}{(x+2)(x-2)})

Cancel out common factors:

(\frac{(x-2)}{(x^2-2x+4)} \times \frac{(x-6)}{1})

So, the simplified expression is:

(\frac{(x-2)(x-6)}{(x^2-2x+4)})

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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