Asked by Shane Agnello on Mar 10, 2024

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Divide and simplify. x2−y22x2−16x÷(x−y) 22xy\frac { x ^ { 2 } - y ^ { 2 } } { 2 x ^ { 2 } - 16 x } \div \frac { ( x - y ) ^ { 2 } } { 2 x y }2x216xx2y2÷2xy(xy) 2

A) y(x−y) 232x2,x≠0\frac { y ( x - y ) ^ { 2 } } { 32 x ^ { 2 } } , x \neq 032x2y(xy) 2,x=0
B) x(x−y) 4(x+y) ,x≠0\frac { x ( x - y ) } { 4 ( x + y ) } , x \neq 04(x+y) x(xy) ,x=0
C) y(x+y) (x−8) (x−y) ,x≠0\frac { y ( x + y ) } { ( x - 8 ) ( x - y ) } , x \neq 0(x8) (xy) y(x+y) ,x=0
D) yx−8,x≠0\frac { y } { x - 8 } , x \neq 0x8y,x=0
E) yx2−64,x≠0\frac { y } { x ^ { 2 } - 64 } , x \neq 0x264y,x=0

Divide

A fundamental arithmetic operation that consists of determining how many times one number is contained within another.

Simplify

To reduce a mathematical expression to its most basic form, making it easier to work with.

  • Undertake multiplication and division activities with rational expressions.
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Verified Answer

TP
Terry Portland

Mar 10, 2024

Final Answer :
C
Explanation :
To divide fractions, we multiply by the reciprocal of the second fraction.

x2−y22x2−16x⋅2xy(x−y)2\frac { x ^ { 2 } - y ^ { 2 } } { 2 x ^ { 2 } - 16 x } \cdot \frac { 2 x y } { ( x - y ) ^ { 2 } }2x216xx2y2(xy)22xy

Now we can factor the numerator and denominator:

(x−y)(x+y)2x(x−8)⋅2xy(x−y)2\frac { (x-y)(x+y) } { 2x(x-8) } \cdot \frac { 2xy } { (x-y)^2 }2x(x8)(xy)(x+y)(xy)22xy

Simplify by canceling out the factors of $(x-y)$:

y(x+y)2x(x−8)\frac { y(x+y) } { 2x(x-8) }2x(x8)y(x+y)

Simplify further:

y(x+y)2x(x+2)(x−4)\frac { y(x+y) } { 2x(x+2)(x-4) }2x(x+2)(x4)y(x+y)

This matches choice C, so the answer is C.