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<ol class="abc-list"><li data-list-item-id="item-1782632548380-4rau6h5sh" data-list-item-path="item-1782632548380-4rau6h5sh"><div class="list-item-paragraph">Express $\frac{6\sqrt{2}-9}{\sqrt{2}-1}$in the form of $a+b\sqrt{2}$, where&nbsp;$a$ and $b$ are integers.[2]</div></li><li data-list-item-id="item-1782632549572-8jjvl9pod" data-list-item-path="item-1782632549572-8jjvl9pod"><div class="list-item-paragraph">$\frac{6\sqrt{2}-9}{\sqrt{2}-1}$ is a root of the equation $x^2 + px + q = 0$ where&nbsp;$p$ and $q$ are integers. Using the result in part (a), find the value of $p$ and of $q$.[4]</div></li></ol>

Express $\frac{6\sqrt{2}-9}{\sqrt{2}-1}$ in the form of $a+b\sqrt{2}$, where $a$ and $b$ are integers.

$\frac{6\sqrt{2}-9}{\sqrt{2}-1}$ is a root of the equation $x^2+px+q=0$ where $p$ and $q$ are integers. Using the result in part (a), find the value of $p$ and of $q$.

Rationalising Denominators

Equating Rational and Irrational Parts

2025 CGSS PRELIMS P1 Q1

Surds

Without using a calculator, find the value of $a$ and of $b$ such that $\sqrt{a - b\sqrt{6}} = \frac{2\sqrt{3}}{3\sqrt{2} + 2\sqrt{3}}$.[4]

Without using a calculator, find the value of $a$ and of $b$ such that $\sqrt{a - b\sqrt{6}} = \frac{2\sqrt{3}}{3\sqrt{2} + 2\sqrt{3}}$.

2023 Ahmad Ibrahim Sec AM S3 EOY Q2

A square of area $ \left(11 + \sqrt{120} \right)\ \text{cm}^2$ has a length of $ \left(\sqrt{a} + \sqrt{b} \right)\ \text{cm}$, where&nbsp;$a &lt; b$.
Without using a calculator, find the values of $a$ and $b$.[5]

A square of area $(11 + \sqrt{120})\text{ cm}^2$ has a length of $(\sqrt{a} + \sqrt{b})\text{ cm}$, where $a < b$.

Without using a calculator, find the values of $a$ and $b$.

Applications of Surds

Without using a calculator, find the value of ${a}$ and of ${b}$ such that $\sqrt{a-b\sqrt{6}}=\frac{2\sqrt{3}}{3\sqrt{2}+2\sqrt{3}}$.[4]

Without using a calculator, find the value of ${a}$ and of ${b}$ such that $\sqrt{a-b\sqrt{6}}=\frac{2\sqrt{3}}{3\sqrt{2}+2\sqrt{3}}$.

2023 AISS AM S3 EOY Q2

Given that $a$ and $b$ are integers, find $a$ and $b$ in the equation $\frac{2-3\sqrt{5}}{a-b\sqrt{5}}=9-4\sqrt{5}$.

Given that $a$ and $b$ are integers, find $a$ and $b$ in the equation $\frac{2-3\sqrt{5}}{a-b\sqrt{5}}=9-4\sqrt{5}$.

It is given that $a$ and $b$ are integers and that $\left( \,1+\sqrt{3}\, \right)\,\,\left( \,a\sqrt{3}-5\, \right)\,\,=\,\,b-7\sqrt{3}\,$.Form a pair of simultaneous equations and solve them to find $a$ and $b$.

It is given that $a$ and $b$ are integers and that $\left( \,1+\sqrt{3}\, \right)\,\,\left( \,a\sqrt{3}-5\, \right)\,\,=\,\,b-7\sqrt{3}\,$.Form a pair of simultaneous equations and solve them to find $a$ and $b$.

It is given that $a$ and $b$ are positive integers such that $\left( a\sqrt{5}-1 \right)\left( \sqrt{5}+b \right)=20\sqrt{5}+32$. Form a pair of simultaneous equations and solve them to find the value of $a$ and of $b$.

It is given that $a$ and $b$ are positive integers such that $\left( a\sqrt{5}-1 \right)\left( \sqrt{5}+b \right)=20\sqrt{5}+32$. Form a pair of simultaneous equations and solve them to find the value of $a$ and of $b$.

It is given that&nbsp;$a$ and $b$ are positive integers such that&nbsp;$a\left( 4+b\sqrt{7} \right)-\left( \sqrt{7}+b \right)=15\sqrt{7}+30$.
Form a pair of simultaneous equations and solve them to find the values of $a$ and $b$.

It is given that $a$ and $b$ are positive integers such that $a\left( 4+b\sqrt{7} \right)-\left( \sqrt{7}+b \right)=15\sqrt{7}+30$.

Form a pair of simultaneous equations and solve them to find the values of $a$ and $b$.

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