Simplify ((6k^3)/(k^2-4))÷((28k^7)/(-6k+12))k/(14k+28)

$\frac{6 k^{3}}{k^{2} - 4} \div \frac{28 k^{7}}{- 6 k + 12} \cdot \frac{k}{14 k + 28}$

Step 1

Rewrite the division as a fraction.

$\frac{\frac{6 k^{3}}{k^{2} - 4}}{(\frac{28 k^{7}}{- 6 k + 12}) (\frac{k}{14 k + 28})}$

Step 2

Cancel the common factor of $28$ and $- 6 k + 12$ .

Factor $2$ out of $28 k^{7}$ .

$\frac{\frac{6 k^{3}}{k^{2} - 4}}{\frac{2 (14 k^{7})}{- 6 k + 12} \cdot \frac{k}{14 k + 28}}$

Cancel the common factors.

Factor $2$ out of $- 6 k$ .

$\frac{\frac{6 k^{3}}{k^{2} - 4}}{\frac{2 (14 k^{7})}{2 (- 3 k) + 12} \cdot \frac{k}{14 k + 28}}$

Factor $2$ out of $12$ .

$\frac{\frac{6 k^{3}}{k^{2} - 4}}{\frac{2 (14 k^{7})}{2 (- 3 k) + 2 (6)} \cdot \frac{k}{14 k + 28}}$

Factor $2$ out of $2 (- 3 k) + 2 (6)$ .

$\frac{\frac{6 k^{3}}{k^{2} - 4}}{\frac{2 (14 k^{7})}{2 (- 3 k + 6)} \cdot \frac{k}{14 k + 28}}$

Cancel the common factor.

$\frac{\frac{6 k^{3}}{k^{2} - 4}}{\frac{2 (14 k^{7})}{2 (- 3 k + 6)} \cdot \frac{k}{14 k + 28}}$

Rewrite the expression.

$\frac{\frac{6 k^{3}}{k^{2} - 4}}{\frac{14 k^{7}}{- 3 k + 6} \cdot \frac{k}{14 k + 28}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{6 k^{3}}{k^{2} - 4} \cdot \frac{1}{\frac{14 k^{7}}{- 3 k + 6} \cdot \frac{k}{14 k + 28}}$

Step 4

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

$\frac{6 k^{3}}{k^{2} - 2^{2}} \cdot \frac{1}{\frac{14 k^{7}}{- 3 k + 6} \cdot \frac{k}{14 k + 28}}$

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

$\frac{6 k^{3}}{(k + 2) (k - 2)} \cdot \frac{1}{\frac{14 k^{7}}{- 3 k + 6} \cdot \frac{k}{14 k + 28}}$

Step 5

Simplify terms.

Combine.

$\frac{6 k^{3} \cdot 1}{(k + 2) (k - 2) (\frac{14 k^{7}}{- 3 k + 6} \cdot \frac{k}{14 k + 28})}$

Factor $3$ out of $- 3 k + 6$ .

Factor $3$ out of $- 3 k$ .

$\frac{6 k^{3} \cdot 1}{(k + 2) (k - 2) (\frac{14 k^{7}}{3 (- k) + 6} \cdot \frac{k}{14 k + 28})}$

Factor $3$ out of $6$ .

$\frac{6 k^{3} \cdot 1}{(k + 2) (k - 2) (\frac{14 k^{7}}{3 (- k) + 3 (2)} \cdot \frac{k}{14 k + 28})}$

Factor $3$ out of $3 (- k) + 3 (2)$ .

$\frac{6 k^{3} \cdot 1}{(k + 2) (k - 2) (\frac{14 k^{7}}{3 (- k + 2)} \cdot \frac{k}{14 k + 28})}$

Factor $14$ out of $14 k + 28$ .

Factor $14$ out of $14 k$ .

$\frac{6 k^{3} \cdot 1}{(k + 2) (k - 2) (\frac{14 k^{7}}{3 (- k + 2)} \cdot \frac{k}{14 (k) + 28})}$

Factor $14$ out of $28$ .

$\frac{6 k^{3} \cdot 1}{(k + 2) (k - 2) (\frac{14 k^{7}}{3 (- k + 2)} \cdot \frac{k}{14 k + 14 \cdot 2})}$

Factor $14$ out of $14 k + 14 \cdot 2$ .

$\frac{6 k^{3} \cdot 1}{(k + 2) (k - 2) (\frac{14 k^{7}}{3 (- k + 2)} \cdot \frac{k}{14 (k + 2)})}$

Multiply $6$ by $1$ .

$\frac{6 k^{3}}{(k + 2) (k - 2) (\frac{14 k^{7}}{3 (- k + 2)} \cdot \frac{k}{14 (k + 2)})}$

Multiply $\frac{14 k^{7}}{3 (- k + 2)}$ by $\frac{k}{14 (k + 2)}$ .

$\frac{6 k^{3}}{(k + 2) (k - 2) \frac{14 k^{7} k}{3 (- k + 2) (14 (k + 2))}}$

Step 6

Simplify the numerator.

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

$\frac{6 k^{3}}{(k + 2) (k - 2) \frac{14 (k^{1} k^{7})}{3 (- k + 2) (14 (k + 2))}}$

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

$\frac{6 k^{3}}{(k + 2) (k - 2) \frac{14 k^{1 + 7}}{3 (- k + 2) (14 (k + 2))}}$

Add $1$ and $7$ .

$\frac{6 k^{3}}{(k + 2) (k - 2) \frac{14 k^{8}}{3 (- k + 2) (14 (k + 2))}}$

Step 7

Reduce the expression by cancelling the common factors.

Multiply $14$ by $3$ .

$\frac{6 k^{3}}{(k + 2) (k - 2) \frac{14 k^{8}}{42 (- k + 2) (k + 2)}}$

Reduce the expression $\frac{14 k^{8}}{42 (- k + 2) (k + 2)}$ by cancelling the common factors.

Factor $14$ out of $14 k^{8}$ .

$\frac{6 k^{3}}{(k + 2) (k - 2) \frac{14 (k^{8})}{42 (- k + 2) (k + 2)}}$

Factor $14$ out of $42 (- k + 2) (k + 2)$ .

$\frac{6 k^{3}}{(k + 2) (k - 2) \frac{14 (k^{8})}{14 (3 (- k + 2) (k + 2))}}$

Cancel the common factor.

$\frac{6 k^{3}}{(k + 2) (k - 2) \frac{14 k^{8}}{14 (3 (- k + 2) (k + 2))}}$

Rewrite the expression.

$\frac{6 k^{3}}{(k + 2) (k - 2) \frac{k^{8}}{3 (- k + 2) (k + 2)}}$

Step 8

Factor $\frac{k^{8}}{3 (- k + 2) (k + 2)}$ out of $\frac{6 k^{3}}{(k + 2) (k - 2) \frac{k^{8}}{3 (- k + 2) (k + 2)}}$ .

$\frac{3 (- k + 2) (k + 2)}{k^{8}} \cdot \frac{6 k^{3}}{(k + 2) (k - 2)}$

Step 9

Simplify terms.

Combine.

$\frac{3 (- k + 2) (k + 2) (6 k^{3})}{k^{8} ((k + 2) (k - 2))}$

Cancel the common factor of $k + 2$ .

Cancel the common factor.

$\frac{3 (- k + 2) (k + 2) (6 k^{3})}{k^{8} ((k + 2) (k - 2))}$

Rewrite the expression.

$\frac{3 (- k + 2) (6 k^{3})}{k^{8} (k - 2)}$

Cancel the common factor of $k^{3}$ and $k^{8}$ .

Factor $k^{3}$ out of $3 (- k + 2) (6 k^{3})$ .

$\frac{k^{3} (3 (- k + 2) \cdot (6))}{k^{8} (k - 2)}$

Cancel the common factors.

Factor $k^{3}$ out of $k^{8} (k - 2)$ .

$\frac{k^{3} (3 (- k + 2) \cdot (6))}{k^{3} (k^{5} (k - 2))}$

Cancel the common factor.

$\frac{k^{3} (3 (- k + 2) \cdot (6))}{k^{3} (k^{5} (k - 2))}$

Rewrite the expression.

$\frac{3 (- k + 2) \cdot (6)}{k^{5} (k - 2)}$

Cancel the common factor of $- k + 2$ and $k - 2$ .

Factor $- 1$ out of $- k$ .

$\frac{3 (- (k) + 2) \cdot 6}{k^{5} (k - 2)}$

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{3 (- (k) - 1 (- 2)) \cdot 6}{k^{5} (k - 2)}$

Factor $- 1$ out of $- (k) - 1 (- 2)$ .

$\frac{3 (- (k - 2)) \cdot 6}{k^{5} (k - 2)}$

Rewrite $- (k - 2)$ as $- 1 (k - 2)$ .

$\frac{3 (- 1 (k - 2)) \cdot 6}{k^{5} (k - 2)}$

Cancel the common factor.

$\frac{3 (- 1 (k - 2)) \cdot 6}{k^{5} (k - 2)}$

Rewrite the expression.

$\frac{3 \cdot (- 1) \cdot 6}{k^{5}}$

Step 10

Simplify the numerator.

Multiply $3$ by $- 1$ .

$\frac{- 3 \cdot 6}{k^{5}}$

Multiply $- 3$ by $6$ .

$\frac{- 18}{k^{5}}$

Step 11

Move the negative in front of the fraction.

$- \frac{18}{k^{5}}$

Simplify ((6k^3)/(k^2-4))÷((28k^7)/(-6k+12))k/(14k+28)

Related Questions