Certainly! Let's go through the process step by step to rationalize the denominator and then perform the subtraction.
### Step 1: Rationalize the denominator.
To rationalize the denominator of the expression [tex]\(\frac{1}{\sqrt{5} + \sqrt{2}}\)[/tex], we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(\sqrt{5} + \sqrt{2}\)[/tex] is [tex]\(\sqrt{5} - \sqrt{2}\)[/tex].
Our expression becomes:
[tex]\[
\frac{1}{\sqrt{5} + \sqrt{2}} \cdot \frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} - \sqrt{2}}
\][/tex]
### Step 2: Multiply out the numerator and the denominator.
First, let's handle the numerator:
[tex]\[
1 \cdot (\sqrt{5} - \sqrt{2}) = \sqrt{5} - \sqrt{2}
\][/tex]
Now, the denominator:
[tex]\[
(\sqrt{5} + \sqrt{2})(\sqrt{5} - \sqrt{2})
\][/tex]
This is a difference of squares:
[tex]\[
(\sqrt{5})^2 - (\sqrt{2})^2 = 5 - 2 = 3
\][/tex]
Putting it all together, the rationalized expression is:
[tex]\[
\frac{\sqrt{5} - \sqrt{2}}{3}
\][/tex]
### Step 3: Perform the subtraction.
We need to subtract the rationalized expression from [tex]\(\sqrt{5} - \sqrt{2}\)[/tex].
So, we have:
[tex]\[
\sqrt{5} - \sqrt{2} - \frac{\sqrt{5} - \sqrt{2}}{3}
\][/tex]
### Step 4: Simplify the expression.
To combine these, we need a common denominator. The common denominator is 3. Rewrite [tex]\(\sqrt{5} - \sqrt{2}\)[/tex] as [tex]\(\frac{3(\sqrt{5} - \sqrt{2})}{3}\)[/tex].
Thus, the expression becomes:
[tex]\[
\frac{3(\sqrt{5} - \sqrt{2})}{3} - \frac{\sqrt{5} - \sqrt{2}}{3}
\][/tex]
Combine the numerators:
[tex]\[
\frac{3(\sqrt{5} - \sqrt{2}) - (\sqrt{5} - \sqrt{2})}{3}
\][/tex]
Factor out [tex]\((\sqrt{5} - \sqrt{2})\)[/tex]:
[tex]\[
\frac{(3 - 1)(\sqrt{5} - \sqrt{2})}{3}
\][/tex]
[tex]\[
\frac{2(\sqrt{5} - \sqrt{2})}{3}
\][/tex]
Thus, the final result after subtracting is:
[tex]\(\frac{2(\sqrt{5} - \sqrt{2})}{3}\)[/tex]
The final expression is:
[tex]\[
-\sqrt{2} - \frac{1}{\sqrt{2} + \sqrt{5}} + \sqrt{5}
\][/tex]
