Asked by Shane Agnello on Mar 10, 2024
Verified
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 }2x2−16xx2−y2÷2xy(x−y) 2
A) y(x−y) 232x2,x≠0\frac { y ( x - y ) ^ { 2 } } { 32 x ^ { 2 } } , x \neq 032x2y(x−y) 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(x−y) ,x=0
C) y(x+y) (x−8) (x−y) ,x≠0\frac { y ( x + y ) } { ( x - 8 ) ( x - y ) } , x \neq 0(x−8) (x−y) y(x+y) ,x=0
D) yx−8,x≠0\frac { y } { x - 8 } , x \neq 0x−8y,x=0
E) yx2−64,x≠0\frac { y } { x ^ { 2 } - 64 } , x \neq 0x2−64y,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.
Verified Answer
TP
Terry PortlandMar 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 } }2x2−16xx2−y2⋅(x−y)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(x−8)(x−y)(x+y)⋅(x−y)22xy
Simplify by canceling out the factors of $(x-y)$:
y(x+y)2x(x−8)\frac { y(x+y) } { 2x(x-8) }2x(x−8)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)(x−4)y(x+y)
This matches choice C, so the answer is C.
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 } }2x2−16xx2−y2⋅(x−y)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(x−8)(x−y)(x+y)⋅(x−y)22xy
Simplify by canceling out the factors of $(x-y)$:
y(x+y)2x(x−8)\frac { y(x+y) } { 2x(x-8) }2x(x−8)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)(x−4)y(x+y)
This matches choice C, so the answer is C.
Learning Objectives
- Undertake multiplication and division activities with rational expressions.