Sure, let's break down the given quotient and simplify it step-by-step.
We start with the quotient:
[tex]\[
\frac{9 + \sqrt{2}}{4 - \sqrt{7}}
\][/tex]
### Step 1: Rationalize the denominator
To eliminate the square root in the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator, which is [tex]\(4 + \sqrt{7}\)[/tex]:
[tex]\[
\frac{(9 + \sqrt{2})(4 + \sqrt{7})}{(4 - \sqrt{7})(4 + \sqrt{7})}
\][/tex]
### Step 2: Expand the numerator and denominator
Let's first handle the numerator:
[tex]\[
(9 + \sqrt{2})(4 + \sqrt{7}) = 9 \cdot 4 + 9 \cdot \sqrt{7} + \sqrt{2} \cdot 4 + \sqrt{2} \cdot \sqrt{7}
\][/tex]
[tex]\[
= 36 + 9\sqrt{7} + 4\sqrt{2} + \sqrt{14}
\][/tex]
Now, let's handle the denominator using the difference of squares:
[tex]\[
(4 - \sqrt{7})(4 + \sqrt{7}) = 4^2 - (\sqrt{7})^2 = 16 - 7 = 9
\][/tex]
So, we have:
[tex]\[
\frac{36 + 9\sqrt{7} + 4\sqrt{2} + \sqrt{14}}{9}
\][/tex]
### Step 3: Simplify the expression
We can now simplify this by dividing each term in the numerator by the denominator:
[tex]\[
\frac{36 + 9\sqrt{7} + 4\sqrt{2} + \sqrt{14}}{9} = \frac{36}{9} + \frac{9\sqrt{7}}{9} + \frac{4\sqrt{2}}{9} + \frac{\sqrt{14}}{9}
\][/tex]
[tex]\[
= 4 + \sqrt{7} + \frac{4\sqrt{2}}{9} + \frac{\sqrt{14}}{9}
\][/tex]
This results in:
[tex]\[
7.690030381760849
\][/tex]
Thus, the final simplified quotient is approximately [tex]\( 7.690030381760849 \)[/tex].
