Asked by Savannah Walters on May 15, 2024
Verified
Multiply and simplify. (x−4y) 2x+4y⋅x2+10xy+24y2x2−16y2\frac { ( x - 4 y ) ^ { 2 } } { x + 4 y } \cdot \frac { x ^ { 2 } + 10 x y + 24 y ^ { 2 } } { x ^ { 2 } - 16 y ^ { 2 } }x+4y(x−4y) 2⋅x2−16y2x2+10xy+24y2
A) (x+4y) (x−6y) x−4y,x≠±4y\frac { ( x + 4 y ) ( x - 6 y ) } { x - 4 y } , x \neq \pm 4 yx−4y(x+4y) (x−6y) ,x=±4y
B) x−4y,x≠−6yx - 4 y , x \neq - 6 yx−4y,x=−6y
C) x+6y,x≠0x + 6 y , x \neq 0x+6y,x=0
D) (x−4y) (x+6y) x+4y,x≠±4y\frac { ( x - 4 y ) ( x + 6 y ) } { x + 4 y } , x \neq \pm 4 yx+4y(x−4y) (x+6y) ,x=±4y
E) (x−6y) (x+6y) x+4y,x≠6y\frac { ( x - 6 y ) ( x + 6 y ) } { x + 4 y } , x \neq 6 yx+4y(x−6y) (x+6y) ,x=6y
Multiply
The mathematical operation of scaling one number by another, representing repeated addition.
Simplify
The process of making an algebraic expression or equation easier to understand by reducing it to its most basic form.
- Conduct operations of multiplication and division on rational exponents.
Verified Answer
WA
Waiza AhmadMay 17, 2024
Final Answer :
D
Explanation :
First, we can simplify the expression inside the first fraction by expanding the square:
(x−4y)2x+4y=x2−8xy+16y2x+4y\frac{(x-4y)^2}{x+4y}=\frac{x^2-8xy+16y^2}{x+4y}x+4y(x−4y)2=x+4yx2−8xy+16y2
Then, we can simplify the expression inside the second fraction by factoring the quadratic:
x2+10xy+24y2x2−16y2=(x+6y)(x+4y)(x+4y)(x−4y)=x+6yx−4y\frac{x^2+10xy+24y^2}{x^2-16y^2}=\frac{(x+6y)(x+4y)}{(x+4y)(x-4y)}=\frac{x+6y}{x-4y}x2−16y2x2+10xy+24y2=(x+4y)(x−4y)(x+6y)(x+4y)=x−4yx+6y
Multiplying the two fractions, we get:
(x2−8xy+16y2)(x+6y)(x+4y)(x−4y)\frac{(x^2-8xy+16y^2)(x+6y)}{(x+4y)(x-4y)}(x+4y)(x−4y)(x2−8xy+16y2)(x+6y)
We can further simplify by factoring the numerator:
(x−4y)(x−6y)(x+6y)(x+4y)(x−4y)\frac{(x-4y)(x-6y)(x+6y)}{(x+4y)(x-4y)}(x+4y)(x−4y)(x−4y)(x−6y)(x+6y)
Canceling out the common factor of $x-4y$, we are left with:
(x−6y)(x+6y)x+4y\frac{(x-6y)(x+6y)}{x+4y}x+4y(x−6y)(x+6y)
Therefore, the final answer is: D.
(x−4y)2x+4y=x2−8xy+16y2x+4y\frac{(x-4y)^2}{x+4y}=\frac{x^2-8xy+16y^2}{x+4y}x+4y(x−4y)2=x+4yx2−8xy+16y2
Then, we can simplify the expression inside the second fraction by factoring the quadratic:
x2+10xy+24y2x2−16y2=(x+6y)(x+4y)(x+4y)(x−4y)=x+6yx−4y\frac{x^2+10xy+24y^2}{x^2-16y^2}=\frac{(x+6y)(x+4y)}{(x+4y)(x-4y)}=\frac{x+6y}{x-4y}x2−16y2x2+10xy+24y2=(x+4y)(x−4y)(x+6y)(x+4y)=x−4yx+6y
Multiplying the two fractions, we get:
(x2−8xy+16y2)(x+6y)(x+4y)(x−4y)\frac{(x^2-8xy+16y^2)(x+6y)}{(x+4y)(x-4y)}(x+4y)(x−4y)(x2−8xy+16y2)(x+6y)
We can further simplify by factoring the numerator:
(x−4y)(x−6y)(x+6y)(x+4y)(x−4y)\frac{(x-4y)(x-6y)(x+6y)}{(x+4y)(x-4y)}(x+4y)(x−4y)(x−4y)(x−6y)(x+6y)
Canceling out the common factor of $x-4y$, we are left with:
(x−6y)(x+6y)x+4y\frac{(x-6y)(x+6y)}{x+4y}x+4y(x−6y)(x+6y)
Therefore, the final answer is: D.
Learning Objectives
- Conduct operations of multiplication and division on rational exponents.