Solved on Feb 21, 2024

Simplify the expression $\frac{x-2}{4 x^{3}} \cdot \frac{x^{2}-1}{x^{2}-x-2}$ .

STEP 1

Assumptions
1. We are given the expression $\frac{x-2}{4 x^{3}} \cdot \frac{x^{2}-1}{x^{2}-x-2}$ .
2. We need to perform the multiplication of the two fractions.
3. We need to express the result in simplest form.
4. We assume $x$ is a variable that does not make any denominator zero.

STEP 2

First, factor the numerators and denominators where possible.
The numerator $x^2 - 1$ is a difference of squares and can be factored as $(x+1)(x-1)$ .
The denominator $x^2 - x - 2$ can be factored as $(x-2)(x+1)$ .

STEP 3

Rewrite the expression with factored terms.
$\frac{x-2}{4 x^{3}} \cdot \frac{(x+1)(x-1)}{(x-2)(x+1)}$

STEP 4

Before multiplying, cancel out any common factors in the numerator and the denominator.
The $(x-2)$ terms cancel out, and one $(x+1)$ term cancels out.

STEP 5

The expression now simplifies to:
$\frac{1}{4 x^{3}} \cdot \frac{x-1}{1}$

STEP 6

Now perform the multiplication of the remaining terms.
$\frac{1 \cdot (x-1)}{4 x^{3} \cdot 1}$

STEP 7

Simplify the expression by multiplying the numerators and denominators.
$\frac{x-1}{4 x^{3}}$

STEP 8

The expression is now in its simplest form.
The final answer is $\frac{x-1}{4 x^{3}}$ .

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Simplify the expression $\frac{x-2}{4 x^{3}} \cdot \frac{x^{2}-1}{x^{2}-x-2}$ .

STEP 1

Assumptions
1. We are given the expression $\frac{x-2}{4 x^{3}} \cdot \frac{x^{2}-1}{x^{2}-x-2}$ .
2. We need to perform the multiplication of the two fractions.
3. We need to express the result in simplest form.
4. We assume $x$ is a variable that does not make any denominator zero.

STEP 2

First, factor the numerators and denominators where possible.
The numerator $x^2 - 1$ is a difference of squares and can be factored as $(x+1)(x-1)$ .
The denominator $x^2 - x - 2$ can be factored as $(x-2)(x+1)$ .

STEP 3

Rewrite the expression with factored terms.
$\frac{x-2}{4 x^{3}} \cdot \frac{(x+1)(x-1)}{(x-2)(x+1)}$

STEP 4

Before multiplying, cancel out any common factors in the numerator and the denominator.
The $(x-2)$ terms cancel out, and one $(x+1)$ term cancels out.

STEP 5

The expression now simplifies to:
$\frac{1}{4 x^{3}} \cdot \frac{x-1}{1}$

STEP 6

Now perform the multiplication of the remaining terms.
$\frac{1 \cdot (x-1)}{4 x^{3} \cdot 1}$

STEP 7

Simplify the expression by multiplying the numerators and denominators.
$\frac{x-1}{4 x^{3}}$

STEP 8

The expression is now in its simplest form.
The final answer is $\frac{x-1}{4 x^{3}}$ .

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Simplify the expression x−24x3⋅x2−1x2−x−2\frac{x-2}{4 x^{3}} \cdot \frac{x^{2}-1}{x^{2}-x-2}4x3x−2​⋅x2−x−2x2−1​.

STEP 1

STEP 2

First, factor the numerators and denominators where possible.The numerator x2−1x^2 - 1x2−1 is a difference of squares and can be factored as (x+1)(x−1)(x+1)(x-1)(x+1)(x−1).The denominator x2−x−2x^2 - x - 2x2−x−2 can be factored as (x−2)(x+1)(x-2)(x+1)(x−2)(x+1).

STEP 3

Rewrite the expression with factored terms.x−24x3⋅(x+1)(x−1)(x−2)(x+1) \frac{x-2}{4 x^{3}} \cdot \frac{(x+1)(x-1)}{(x-2)(x+1)} 4x3x−2​⋅(x−2)(x+1)(x+1)(x−1)​

STEP 4

Before multiplying, cancel out any common factors in the numerator and the denominator.The (x−2)(x-2)(x−2) terms cancel out, and one (x+1)(x+1)(x+1) term cancels out.

STEP 5

The expression now simplifies to:14x3⋅x−11 \frac{1}{4 x^{3}} \cdot \frac{x-1}{1} 4x31​⋅1x−1​

STEP 6

Now perform the multiplication of the remaining terms.1⋅(x−1)4x3⋅1 \frac{1 \cdot (x-1)}{4 x^{3} \cdot 1} 4x3⋅11⋅(x−1)​

STEP 7

Simplify the expression by multiplying the numerators and denominators.x−14x3 \frac{x-1}{4 x^{3}} 4x3x−1​

STEP 8

The expression is now in its simplest form.The final answer is x−14x3\frac{x-1}{4 x^{3}}4x3x−1​.

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Simplify the expression $\frac{x-2}{4 x^{3}} \cdot \frac{x^{2}-1}{x^{2}-x-2}$ .

First, factor the numerators and denominators where possible.
The numerator $x^2 - 1$ is a difference of squares and can be factored as $(x+1)(x-1)$ .
The denominator $x^2 - x - 2$ can be factored as $(x-2)(x+1)$ .

Rewrite the expression with factored terms.
$\frac{x-2}{4 x^{3}} \cdot \frac{(x+1)(x-1)}{(x-2)(x+1)}$

Before multiplying, cancel out any common factors in the numerator and the denominator.
The $(x-2)$ terms cancel out, and one $(x+1)$ term cancels out.

The expression now simplifies to:
$\frac{1}{4 x^{3}} \cdot \frac{x-1}{1}$

Now perform the multiplication of the remaining terms.
$\frac{1 \cdot (x-1)}{4 x^{3} \cdot 1}$

Simplify the expression by multiplying the numerators and denominators.
$\frac{x-1}{4 x^{3}}$

The expression is now in its simplest form.
The final answer is $\frac{x-1}{4 x^{3}}$ .