Sure, let's work through each part of the question step-by-step.
### Part (a)
We are asked to write [tex]\(\frac{6}{\sqrt{5} - \sqrt{2}}\)[/tex] in the form [tex]\(a \sqrt{5} + b \sqrt{2}\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are integers.
First, we need to rationalize the denominator. To do this, we multiply the numerator and the denominator by the conjugate of the denominator, which is [tex]\(\sqrt{5} + \sqrt{2}\)[/tex]:
[tex]\[
\frac{6}{\sqrt{5} - \sqrt{2}} = \frac{6 (\sqrt{5} + \sqrt{2})}{(\sqrt{5} - \sqrt{2})(\sqrt{5} + \sqrt{2})}
\][/tex]
Now, let's simplify the denominator:
[tex]\[
(\sqrt{5} - \sqrt{2})(\sqrt{5} + \sqrt{2}) = \sqrt{5}^2 - \sqrt{2}^2 = 5 - 2 = 3
\][/tex]
So the expression becomes:
[tex]\[
\frac{6 (\sqrt{5} + \sqrt{2})}{3}
\][/tex]
Now, simplify the numerator:
[tex]\[
\frac{6 (\sqrt{5} + \sqrt{2})}{3} = 2 (\sqrt{5} + \sqrt{2}) = 2 \sqrt{5} + 2 \sqrt{2}
\][/tex]
Thus, [tex]\(\frac{6}{\sqrt{5} - \sqrt{2}}\)[/tex] can be written in the form [tex]\(a \sqrt{5} + b \sqrt{2}\)[/tex] where [tex]\(a = 2\)[/tex] and [tex]\(b = 2\)[/tex].
### Part (b)
Now, we need to solve the equation
[tex]\[
\sqrt{5} x = \sqrt{2} x + 18 \sqrt{5}
\][/tex]
First, let's isolate the terms involving [tex]\(x\)[/tex]. Subtract [tex]\(\sqrt{2} x\)[/tex] from both sides of the equation:
[tex]\[
\sqrt{5} x - \sqrt{2} x = 18 \sqrt{5}
\][/tex]
Factor out [tex]\(x\)[/tex] on the left side:
[tex]\[
x (\sqrt{5} - \sqrt{2}) = 18 \sqrt{5}
\][/tex]
To solve for [tex]\(x\)[/tex], divide both sides by [tex]\(\sqrt{5} - \sqrt{2}\)[/tex]:
[tex]\[
x = \frac{18 \sqrt{5}}{\sqrt{5} - \sqrt{2}}
\][/tex]
We already rationalized a similar expression in part (a), where we found that:
[tex]\[
\frac{6}{\sqrt{5} - \sqrt{2}} = 2 \sqrt{5} + 2 \sqrt{2}
\][/tex]
So, we can adjust this result to find the equivalent form for our current expression. Notice:
[tex]\[
\frac{18 \sqrt{5}}{\sqrt{5} - \sqrt{2}} = 3 \cdot \frac{6 \sqrt{5}}{\sqrt{5} - \sqrt{2}} = 3 \left( 2 \sqrt{5} + 2 \sqrt{2} \right)
\][/tex]
Thus, we get:
[tex]\[
x = 3 \left( 2 \sqrt{5} + 2 \sqrt{2} \right)
\][/tex]
Simplifying this:
[tex]\[
x = 6 \sqrt{5} + 6 \sqrt{2}
\][/tex]
Therefore, the solution to the equation [tex]\(\sqrt{5} x = \sqrt{2} x + 18 \sqrt{5}\)[/tex] is:
[tex]\[
x = 6 \sqrt{5} + 6 \sqrt{2}
\][/tex]
In its simplest form.
