Simplify 3/(x-10)(x^2-100)/(2x)(x+7)/(x^3+17x^2+70x)

$\frac{3}{x - 10} \cdot \frac{x^{2} - 100}{2 x} \cdot \frac{x + 7}{x^{3} + 17 x^{2} + 70 x}$

Simplify the denominator.

Factor $x$ out of $x^{3} + 17 x^{2} + 70 x$ .

Factor $x$ out of $x^{3}$ .

$\frac{3}{x - 10} \cdot \frac{x^{2} - 100}{2 x} \cdot \frac{x + 7}{x \cdot x^{2} + 17 x^{2} + 70 x}$

Factor $x$ out of $17 x^{2}$ .

$\frac{3}{x - 10} \cdot \frac{x^{2} - 100}{2 x} \cdot \frac{x + 7}{x \cdot x^{2} + x (17 x) + 70 x}$

Factor $x$ out of $70 x$ .

$\frac{3}{x - 10} \cdot \frac{x^{2} - 100}{2 x} \cdot \frac{x + 7}{x \cdot x^{2} + x (17 x) + x \cdot 70}$

Factor $x$ out of $x \cdot x^{2} + x (17 x)$ .

$\frac{3}{x - 10} \cdot \frac{x^{2} - 100}{2 x} \cdot \frac{x + 7}{x (x^{2} + 17 x) + x \cdot 70}$

Factor $x$ out of $x (x^{2} + 17 x) + x \cdot 70$ .

$\frac{3}{x - 10} \cdot \frac{x^{2} - 100}{2 x} \cdot \frac{x + 7}{x (x^{2} + 17 x + 70)}$

Factor $x^{2} + 17 x + 70$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $70$ and whose sum is $17$ .

$7, 10$

Write the factored form using these integers.

$\frac{3}{x - 10} \cdot \frac{x^{2} - 100}{2 x} \cdot \frac{x + 7}{x ((x + 7) (x + 10))}$

$\frac{3}{x - 10} \cdot \frac{x^{2} - 100}{2 x} \cdot \frac{x + 7}{x (x + 7) (x + 10)}$

Simplify the numerator.

Rewrite $100$ as $10^{2}$ .

$\frac{3}{x - 10} \cdot \frac{x^{2} - 10^{2}}{2 x} \cdot \frac{x + 7}{x (x + 7) (x + 10)}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 10$ .

$\frac{3}{x - 10} \cdot \frac{(x + 10) (x - 10)}{2 x} \cdot \frac{x + 7}{x (x + 7) (x + 10)}$

Simplify terms.

Cancel the common factor of $x - 10$ .

Factor $x - 10$ out of $(x + 10) (x - 10)$ .

$\frac{3}{x - 10} \cdot \frac{(x - 10) (x + 10)}{2 x} \cdot \frac{x + 7}{x (x + 7) (x + 10)}$

Cancel the common factor.

$\frac{3}{x - 10} \cdot \frac{(x - 10) (x + 10)}{2 x} \cdot \frac{x + 7}{x (x + 7) (x + 10)}$

Rewrite the expression.

$3 \cdot \frac{x + 10}{2 x} \cdot \frac{x + 7}{x (x + 7) (x + 10)}$

Combine $3$ and $\frac{x + 10}{2 x}$ .

$\frac{3 (x + 10)}{2 x} \cdot \frac{x + 7}{x (x + 7) (x + 10)}$

Cancel the common factor of $x + 10$ .

Factor $x + 10$ out of $3 (x + 10)$ .

$\frac{(x + 10) \cdot 3}{2 x} \cdot \frac{x + 7}{x (x + 7) (x + 10)}$

Factor $x + 10$ out of $x (x + 7) (x + 10)$ .

$\frac{(x + 10) \cdot 3}{2 x} \cdot \frac{x + 7}{(x + 10) (x (x + 7))}$

Cancel the common factor.

$\frac{(x + 10) \cdot 3}{2 x} \cdot \frac{x + 7}{(x + 10) (x (x + 7))}$

Rewrite the expression.

$\frac{3}{2 x} \cdot \frac{x + 7}{x (x + 7)}$

Multiply $\frac{3}{2 x}$ by $\frac{x + 7}{x (x + 7)}$ .

$\frac{3 (x + 7)}{2 x (x (x + 7))}$

Raise $x$ to the power of $1$ .

$\frac{3 (x + 7)}{2 (x^{1} x) (x + 7)}$

Raise $x$ to the power of $1$ .

$\frac{3 (x + 7)}{2 (x^{1} x^{1}) (x + 7)}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 (x + 7)}{2 x^{1 + 1} (x + 7)}$

Add $1$ and $1$ .

$\frac{3 (x + 7)}{2 x^{2} (x + 7)}$

Cancel the common factor of $x + 7$ .

Cancel the common factor.

$\frac{3 (x + 7)}{2 x^{2} (x + 7)}$

Rewrite the expression.

$\frac{3}{2 x^{2}}$

Simplify 3/(x-10)*(x^2-100)/(2x)*(x+7)/(x^3+17x^2+70x)

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Simplify 3/(x-10)(x^2-100)/(2x)(x+7)/(x^3+17x^2+70x)