Simplify (v^2-49)/(20v^3)*(4v^2-24v)/(v^2+v-42)

$\frac{v^{2} - 49}{20 v^{3}} \cdot \frac{4 v^{2} - 24 v}{v^{2} + v - 42}$

Step 1

Simplify the numerator.

Rewrite $49$ as $7^{2}$ .

$\frac{v^{2} - 7^{2}}{20 v^{3}} \cdot \frac{4 v^{2} - 24 v}{v^{2} + v - 42}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = v$ and $b = 7$ .

$\frac{(v + 7) (v - 7)}{20 v^{3}} \cdot \frac{4 v^{2} - 24 v}{v^{2} + v - 42}$

Step 2

Factor $4 v$ out of $4 v^{2} - 24 v$ .

Factor $4 v$ out of $4 v^{2}$ .

$\frac{(v + 7) (v - 7)}{20 v^{3}} \cdot \frac{4 v (v) - 24 v}{v^{2} + v - 42}$

Factor $4 v$ out of $- 24 v$ .

$\frac{(v + 7) (v - 7)}{20 v^{3}} \cdot \frac{4 v (v) + 4 v (- 6)}{v^{2} + v - 42}$

Factor $4 v$ out of $4 v (v) + 4 v (- 6)$ .

$\frac{(v + 7) (v - 7)}{20 v^{3}} \cdot \frac{4 v (v - 6)}{v^{2} + v - 42}$

Step 3

Factor $v^{2} + v - 42$ 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 $- 42$ and whose sum is $1$ .

$- 6, 7$

Write the factored form using these integers.

$\frac{(v + 7) (v - 7)}{20 v^{3}} \cdot \frac{4 v (v - 6)}{(v - 6) (v + 7)}$

Step 4

Simplify terms.

Combine.

$\frac{(v + 7) (v - 7) (4 v (v - 6))}{20 v^{3} ((v - 6) (v + 7))}$

Cancel the common factor of $v + 7$ .

Cancel the common factor.

$\frac{(v + 7) (v - 7) (4 v (v - 6))}{20 v^{3} ((v - 6) (v + 7))}$

Rewrite the expression.

$\frac{(v - 7) (4 v (v - 6))}{20 v^{3} (v - 6)}$

Cancel the common factor of $4$ and $20$ .

Factor $4$ out of $(v - 7) (4 v (v - 6))$ .

$\frac{4 ((v - 7) (v (v - 6)))}{20 v^{3} (v - 6)}$

Cancel the common factors.

Factor $4$ out of $20 v^{3} (v - 6)$ .

$\frac{4 ((v - 7) (v (v - 6)))}{4 (5 v^{3} (v - 6))}$

Cancel the common factor.

$\frac{4 ((v - 7) (v (v - 6)))}{4 (5 v^{3} (v - 6))}$

Rewrite the expression.

$\frac{(v - 7) (v (v - 6))}{5 v^{3} (v - 6)}$

Cancel the common factor of $v$ and $v^{3}$ .

Factor $v$ out of $(v - 7) (v (v - 6))$ .

$\frac{v ((v - 7) (v - 6))}{5 v^{3} (v - 6)}$

Cancel the common factors.

Factor $v$ out of $5 v^{3} (v - 6)$ .

$\frac{v ((v - 7) (v - 6))}{v (5 v^{2} (v - 6))}$

Cancel the common factor.

$\frac{v ((v - 7) (v - 6))}{v (5 v^{2} (v - 6))}$

Rewrite the expression.

$\frac{(v - 7) (v - 6)}{5 v^{2} (v - 6)}$

Cancel the common factor of $v - 6$ .

Cancel the common factor.

$\frac{(v - 7) (v - 6)}{5 v^{2} (v - 6)}$

Rewrite the expression.

$\frac{v - 7}{5 v^{2}}$

Simplify (v^2-49)/(20v^3)*(4v^2-24v)/(v^2+v-42)

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