Certainly! Let's solve each part of the given question step-by-step.
### (a) Factorise [tex]\( rs + tr - pt - ps \)[/tex]:
1. Identify common terms and group them:
[tex]\[
rs + tr - pt - ps
\][/tex]
Group the terms:
[tex]\[
(rs + tr) + (-pt - ps)
\][/tex]
2. Factor out the common factors in each group:
From the first group [tex]\( (rs + tr) \)[/tex], factor out [tex]\( r \)[/tex]:
[tex]\[
r(s + t)
\][/tex]
From the second group [tex]\((-pt - ps)\)[/tex], factor out [tex]\(-p \)[/tex]:
[tex]\[
-p(t + s)
\][/tex]
3. Rewriting the expression with factored groups:
[tex]\[
r(s + t) - p(t + s)
\][/tex]
4. Notice that [tex]\((s + t)\)[/tex] is a common factor in both terms:
Factor out [tex]\((s + t)\)[/tex]:
[tex]\[
(r - p)(s + t)
\][/tex]
Thus, the factorised form of the expression [tex]\( rs + tr - pt - ps \)[/tex] is:
[tex]\[
\boxed{-(p - r)(s + t)}
\][/tex]
### (b) Rationalize [tex]\(\frac{1}{\sqrt{2} + \sqrt{5}}\)[/tex]:
1. Multiply the numerator and the denominator by the conjugate of the denominator:
The conjugate of [tex]\(\sqrt{2} + \sqrt{5}\)[/tex] is [tex]\(\sqrt{2} - \sqrt{5}\)[/tex].
[tex]\[
\frac{1}{\sqrt{2} + \sqrt{5}} \times \frac{\sqrt{2} - \sqrt{5}}{\sqrt{2} - \sqrt{5}}
\][/tex]
2. Multiply the expressions:
In the numerator:
[tex]\[
1 \times (\sqrt{2} - \sqrt{5}) = \sqrt{2} - \sqrt{5}
\][/tex]
In the denominator, apply the difference of squares formula [tex]\( (a + b)(a - b) = a^2 - b^2 \)[/tex]:
[tex]\[
(\sqrt{2} + \sqrt{5})(\sqrt{2} - \sqrt{5}) = (\sqrt{2})^2 - (\sqrt{5})^2 = 2 - 5 = -3
\][/tex]
3. Write the rationalized expression:
[tex]\[
\frac{\sqrt{2} - \sqrt{5}}{2 - 5}
\][/tex]
Simplify the denominator [tex]\( -3 \)[/tex]:
[tex]\[
\frac{\sqrt{2} - \sqrt{5}}{-3} = \frac{1}{-\sqrt{2} + \sqrt{5}}
\][/tex]
Thus, the rationalized form of [tex]\(\frac{1}{\sqrt{2} + \sqrt{5}}\)[/tex] is:
[tex]\[
\boxed{\frac{1}{-(\sqrt{2} - \sqrt{5})}}
\][/tex]
In conclusion:
- The factorized form of [tex]\( rs + tr - pt - ps \)[/tex] is [tex]\(-(p - r)(s + t)\)[/tex].
- The rationalized form of [tex]\(\frac{1}{\sqrt{2} + \sqrt{5}}\)[/tex] is [tex]\(\frac{1}{-(\sqrt{2} - \sqrt{5})}\)[/tex].
