Asked by Yvonne Manning on Apr 27, 2024

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Rationalize the denominator of y7−y\frac { \sqrt { y } } { 7 - \sqrt { y } }7yy and simplify.

A) y\sqrt { y }y
B) 7+y49−y\frac { 7 + y } { 49 - y }49y7+y
C) 7y−y49−y\frac { 7 \sqrt { y } - y } { 49 - y }49y7yy
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 }49y7y+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.
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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*}7yy=(7y)(7+y)y(7+y)=49yy(7+y)=49y7y+y
Therefore, the best choice is $\boxed{\text{(E)}}$.