Asked by Nitin Channa on May 06, 2024
Verified
Simplify the expression. 9x−1+9y−1x−1−y−1\frac { 9 x ^ { - 1 } + 9 y ^ { - 1 } } { x ^ { - 1 } - y ^ { - 1 } }x−1−y−19x−1+9y−1
A) −18,x≠0,y≠0- 18 , x \neq 0 , y \neq 0−18,x=0,y=0
B) x+y9xy,x≠0,y≠0\frac { x + y } { 9 x y } , x \neq 0 , y \neq 09xyx+y,x=0,y=0
C) xy9(x+y) ,x≠0,y≠0\frac { x y } { 9 ( x + y ) } , x \neq 0 , y \neq 09(x+y) xy,x=0,y=0
D) 9(x+y) y−x,x≠0,y≠0\frac { 9 ( x + y ) } { y - x } , x \neq 0 , y \neq 0y−x9(x+y) ,x=0,y=0
E) 9(x−y) xy,x≠0,y≠0\frac { 9 ( x - y ) } { x y } , x \neq 0 , y \neq 0xy9(x−y) ,x=0,y=0
Expression
A combination of symbols and numbers that, together, represent a value or concept.
Simplify
Simplifying in mathematics involves altering an expression or equation to make it easier to work with, often by combining like terms and reducing expressions to their simplest form.
- Simplify algebraic expressions that contain variables and exponents.
Verified Answer
CA
Clarissa AurelliaMay 10, 2024
Final Answer :
D
Explanation :
To simplify the expression, we can first cross-multiply the denominators:
9x−1+9y−1x−1−y−1=9x−1+9y−1(xy)−1(x−1−y−1)=9x−1+9y−1y−xxy=9x−1+9y−1−1xy(x−y)=9x−1+9y−11xy(y−x)=9(x−1+y−1)y−x\begin{align*}\frac { 9 x ^ { - 1 } + 9 y ^ { - 1 } } { x ^ { - 1 } - y ^ { - 1 } } &= \frac{9x^{-1}+9y^{-1}}{(xy)^{-1}(x^{-1}-y^{-1})}\\&=\frac{9x^{-1}+9y^{-1}}{\frac{y-x}{xy}}\\&=\frac{9x^{-1}+9y^{-1}}{\frac{-1}{xy}(x-y)}\\&=\frac{9x^{-1}+9y^{-1}}{\frac{1}{xy}(y-x)}\\&=\frac{9(x^{-1}+y^{-1})}{y-x}\end{align*}x−1−y−19x−1+9y−1=(xy)−1(x−1−y−1)9x−1+9y−1=xyy−x9x−1+9y−1=xy−1(x−y)9x−1+9y−1=xy1(y−x)9x−1+9y−1=y−x9(x−1+y−1)
Therefore, the correct answer is D.
9x−1+9y−1x−1−y−1=9x−1+9y−1(xy)−1(x−1−y−1)=9x−1+9y−1y−xxy=9x−1+9y−1−1xy(x−y)=9x−1+9y−11xy(y−x)=9(x−1+y−1)y−x\begin{align*}\frac { 9 x ^ { - 1 } + 9 y ^ { - 1 } } { x ^ { - 1 } - y ^ { - 1 } } &= \frac{9x^{-1}+9y^{-1}}{(xy)^{-1}(x^{-1}-y^{-1})}\\&=\frac{9x^{-1}+9y^{-1}}{\frac{y-x}{xy}}\\&=\frac{9x^{-1}+9y^{-1}}{\frac{-1}{xy}(x-y)}\\&=\frac{9x^{-1}+9y^{-1}}{\frac{1}{xy}(y-x)}\\&=\frac{9(x^{-1}+y^{-1})}{y-x}\end{align*}x−1−y−19x−1+9y−1=(xy)−1(x−1−y−1)9x−1+9y−1=xyy−x9x−1+9y−1=xy−1(x−y)9x−1+9y−1=xy1(y−x)9x−1+9y−1=y−x9(x−1+y−1)
Therefore, the correct answer is D.
Learning Objectives
- Simplify algebraic expressions that contain variables and exponents.
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