Rationalizing the Denominator

To find the value of $y$, we first simplify the given expression by rationalizing the denominator.

The expression is $\frac{2\sqrt{2}+\sqrt{7}}{2\sqrt{2}-\sqrt{7}}$.

Multiply the numerator and the denominator by the conjugate of the denominator, which is $2\sqrt{2}+\sqrt{7}$:

$ \frac{2\sqrt{2}+\sqrt{7}}{2\sqrt{2}-\sqrt{7}} \times \frac{2\sqrt{2}+\sqrt{7}}{2\sqrt{2}+\sqrt{7}} $

Simplifying the Expression

Numerator calculation:

$ (2\sqrt{2}+\sqrt{7})(2\sqrt{2}+\sqrt{7}) = (2\sqrt{2})^2 + 2(2\sqrt{2})(\sqrt{7}) + (\sqrt{7})^2 $

$ = (4 \times 2) + 4\sqrt{14} + 7 $

$ = 8 + 4\sqrt{14} + 7 $

$ = 15 + 4\sqrt{14} $

Denominator calculation:

$ (2\sqrt{2}-\sqrt{7})(2\sqrt{2}+\sqrt{7}) = (2\sqrt{2})^2 - (\sqrt{7})^2 $

$ = (4 \times 2) - 7 $

$ = 8 - 7 $

$ = 1 $

Finding the Value of y

Now, substitute the simplified numerator and denominator back into the expression:

$ \frac{15 + 4\sqrt{14}}{1} = 15 + 4\sqrt{14} $

We are given that the expression equals $x + y\sqrt{14}$. By comparing $15 + 4\sqrt{14}$ with $x + y\sqrt{14}$, we can identify the values of $x$ and $y$.

• $x = 15$
• $y = 4$

Therefore, the value of $y$ is 4.