How do you simplify $\frac{x^{2} - 16 y^{2}}{x^{2} + 3 x y - 4 y^{2}} \div \frac{x^{2} - 8 x y + 16 y^{2}}{x - y}$ $\frac { x ^ { 2} - 16y ^ { 2} } { x ^ { 2} + 3x y - 4y ^ { 2} } \div \frac { x ^ { 2} - 8x y + 16y ^ { 2} } { x - y }$ ?

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How do you simplify $\frac{x^{2} - 16 y^{2}}{x^{2} + 3 x y - 4 y^{2}} \div \frac{x^{2} - 8 x y + 16 y^{2}}{x - y}$ $\frac { x ^ { 2} - 16y ^ { 2} } { x ^ { 2} + 3x y - 4y ^ { 2} } \div \frac { x ^ { 2} - 8x y + 16y ^ { 2} } { x - y }$ ?

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Answer 1 · 2018-05-03T21:21:47

$\frac{1}{x - 4 y}$ $1/ (x - 4y)$

Explanation:

Recall that dividing something by $x$ $x$ is the same as multiplying it by the inverse $\frac{1}{x}$ $1/x$ . That is, $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$ $a/b div c/d = a/b * d/c$ . We use this algebraic fact to help us simplify.

$\frac{x^{2} - 16 y^{2}}{x^{2} + 3 x y - 4 y^{2}} \div \frac{x^{2} - 8 x y + 16 y^{2}}{x - y}$ $(x^2 - 16y^2)/(x^2 + 3xy - 4y^2) div (x^2 - 8xy + 16y^2)/(x-y)$
$= \frac{(x^{2} - 16 y^{2}) (x - y)}{(x^{2} + 3 x y - 4 y^{2}) (x^{2} - 8 x y + 16 y^{2})}$ $= ((x^2 - 16y^2)(x-y))/((x^2 + 3xy - 4y^2)(x^2 - 8xy + 16y^2))$

Now we note that most of these terms can be factored. See that the following are true:

$x^{2} - 16 y^{2} = (x - 4 y) (x + 4 y)$ $x^2 - 16y^2 = (x-4y)(x+4y)$
$x^{2} - 8 x y + 16 y^{2} = (x - 4 y) (x - 4 y)$ $x^2 - 8xy + 16y^2 = (x-4y)(x-4y)$
$x^{2} + 3 x y - 4 y^{2} = (x + 4 y) (x - y)$ $x^2 + 3xy - 4y^2 = (x+4y)(x - y)$

Making the replacements as needed, this gives

$\frac{(x - 4 y) (x + 4 y) (x - y)}{(x + 4 y) (x - y) {(x - 4 y)}^{2}}$ $((x-4y)(x+4y)(x-y))/((x+4y)(x-y)(x-4y)^2)$
$= \frac{1}{x - 4 y} \cdot \frac{x - 4 y}{x - 4 y} \cdot \frac{x + 4 y}{x + 4 y} \cdot \frac{x - y}{x - y}$ $= 1/(x-4y) * (x-4y)/(x-4y) * (x+4y)/(x+4y) * (x-y)/(x-y)$
$= \frac{1}{x - 4 y}$ $= 1 / (x - 4y)$

Thus, our final answer is $\frac{1}{x - 4 y}$ $1 / (x-4y)$ .

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How do you simplify $\frac{x^{2} - 16 y^{2}}{x^{2} + 3 x y - 4 y^{2}} \div \frac{x^{2} - 8 x y + 16 y^{2}}{x - y}$ $\frac { x ^ { 2} - 16y ^ { 2} } { x ^ { 2} + 3x y - 4y ^ { 2} } \div \frac { x ^ { 2} - 8x y + 16y ^ { 2} } { x - y }$ ?

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Explanation:

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How do you simplify x2−16y2x2+3xy−4y2÷x2−8xy+16y2x−y\frac { x ^ { 2} - 16y ^ { 2} } { x ^ { 2} + 3x y - 4y ^ { 2} } \div \frac { x ^ { 2} - 8x y + 16y ^ { 2} } { x - y }?

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Explanation:

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How do you simplify $\frac{x^{2} - 16 y^{2}}{x^{2} + 3 x y - 4 y^{2}} \div \frac{x^{2} - 8 x y + 16 y^{2}}{x - y}$ $\frac { x ^ { 2} - 16y ^ { 2} } { x ^ { 2} + 3x y - 4y ^ { 2} } \div \frac { x ^ { 2} - 8x y + 16y ^ { 2} } { x - y }$ ?