Asked by Yvonne Manning on Apr 27, 2024
Verified
Rationalize the denominator of y7−y\frac { \sqrt { y } } { 7 - \sqrt { y } }7−yy and simplify.
A) y\sqrt { y }y
B) 7+y49−y\frac { 7 + y } { 49 - y }49−y7+y
C) 7y−y49−y\frac { 7 \sqrt { y } - y } { 49 - y }49−y7y−y
D) 7y+y49+y\frac { 7 \sqrt { y } + y } { 49 + y }49+y7y+y
E) 7y+y49−y\frac { 7 \sqrt { y } + y } { 49 - y }49−y7y+y
Rationalize
The process of modifying an expression to eliminate radicals from the denominator or complex numbers from the denominator of a fraction.
Denominator
The bottom part of a fraction that tells into how many equal parts the whole is divided.
- Develop skill in the rationalization of denominators within fractions containing square roots.
Verified Answer
CN
Chizoba NnakweApr 28, 2024
Final Answer :
E
Explanation :
To rationalize the denominator, we need to multiply both the numerator and denominator by the conjugate of the denominator: 7+y7+\sqrt{y}7+y .
y7−y=y⋅(7+y)(7−y)⋅(7+y)=y⋅(7+y)49−y=7y+y49−y\begin{align*}\frac{\sqrt{y}}{7-\sqrt{y}} &= \frac{\sqrt{y}\cdot(7+\sqrt{y})}{(7-\sqrt{y})\cdot(7+\sqrt{y})}\\&= \frac{\sqrt{y}\cdot(7+\sqrt{y})}{49-y}\\&= \frac{7\sqrt{y}+y}{49-y}\end{align*}7−yy=(7−y)⋅(7+y)y⋅(7+y)=49−yy⋅(7+y)=49−y7y+y
Therefore, the best choice is $\boxed{\text{(E)}}$.
y7−y=y⋅(7+y)(7−y)⋅(7+y)=y⋅(7+y)49−y=7y+y49−y\begin{align*}\frac{\sqrt{y}}{7-\sqrt{y}} &= \frac{\sqrt{y}\cdot(7+\sqrt{y})}{(7-\sqrt{y})\cdot(7+\sqrt{y})}\\&= \frac{\sqrt{y}\cdot(7+\sqrt{y})}{49-y}\\&= \frac{7\sqrt{y}+y}{49-y}\end{align*}7−yy=(7−y)⋅(7+y)y⋅(7+y)=49−yy⋅(7+y)=49−y7y+y
Therefore, the best choice is $\boxed{\text{(E)}}$.
Learning Objectives
- Develop skill in the rationalization of denominators within fractions containing square roots.