Rationalising Surds

Q. Question 10
Rationalize the denominator of the following
$\left(i\right)\frac{2}{3\sqrt{3}}\left(ii\right)\frac{\sqrt{40}}{\sqrt{3}}\left(iii\right)\frac{3+\sqrt{2}}{4\sqrt{2}}\left(iv\right)\frac{16}{\sqrt{41-5}}\left(v\right)\frac{2+\sqrt{3}}{2-\sqrt{3}}\left(vi\right)\frac{\sqrt{6}}{\sqrt{2}+\sqrt{3}}\left(vii\right)\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}$
$\left(viii\right)\frac{3\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$
$\left(ix\right)\frac{4\sqrt{3}+5\sqrt{2}}{\sqrt{48}+\sqrt{18}}$
Thinking process
For rationalizing the denominator multiplying numerator by the conjugate of the denominator term and simplify it to get the result.

Q. Question 4
If $a=\frac{3+\sqrt{5}}{2}$, then find the value of $a^2+\frac{1}{a^2}$.

Q.

The rationalisation factor of $2+\sqrt{3}$ is

1. $2-\sqrt{3}$

2. $\sqrt{2}-3$

3. $\sqrt{2}+3$

4. $\sqrt{3}-2$

Q.

Rationalise the denominator of $\frac{1}{\sqrt{3}-\sqrt{2}}$.

1. $\sqrt{3}+\sqrt{2}$

2. $\sqrt{5}+\sqrt{2}$

3. $\sqrt{3}-\sqrt{2}$

4. $\sqrt{5}+\sqrt{3}$

Q. If $\frac{3+\sqrt{5}}{2\sqrt{5}+3}=a+b\sqrt{5}$, then the values of irrational numbers $a$ and $b$ are

1. 
2. 
3. 
4. 

Q.

Evaluate: $\sqrt{12-\sqrt{68+48\sqrt{2}}}$

1. $\sqrt{2}-3$

2. $2+\sqrt{2}$

3. $2-\sqrt{2}$

4. $6–2\sqrt{2}$

Q.

Simplify $\left(\frac{2+\sqrt{5}}{2-\sqrt{5}}\right)+\left(\frac{2-\sqrt{5}}{2+\sqrt{5}}\right)$

1. -18

2. + 18

3. 18 + 8√5

4. 18 - 8√5

Q.

If a and b are rational numbers, find 'a' and 'b' when

$\frac{\sqrt{7}-2}{\sqrt{7}+2}=a\sqrt{7}+b$

1. $a=\frac{4}{3},b=\frac{11}{3}$

2. $a=\frac{-4}{3},b=\frac{11}{3}$

3. $a=-\frac{4}{3},b=\frac{-11}{3}$

4. $a=\frac{4}{3},b=\frac{-11}{3}$

Q. If $x=3+\sqrt{2}$ then the value of $x-\frac{1}{x}$ is rational or irrational

Q. If $\sqrt{2}=1.4142$...... then $\sqrt{\frac{\sqrt{2}-1}{\sqrt{2}+1}}$ is equal to:

1. 2.4142
2. 5.8282
3. 0.4142
4. 0.1718

Q. Solve ($\sqrt{6}$+$\sqrt{3}{)}^2$

Q. Rationalise the denominator of $\frac{1}{\sqrt{6}-\sqrt{5}-\sqrt{11}}$

1. -$\frac{\sqrt{60}(\sqrt{6}+\sqrt{5}+\sqrt{11})}{60}$
2. $\frac{\sqrt{30}(\sqrt{6}+\sqrt{5}+\sqrt{11})}{60}$
3. $\frac{\sqrt{3}(\sqrt{6}+\sqrt{5}+\sqrt{11})}{60}$
4. -$\frac{\sqrt{30}(\sqrt{6}+\sqrt{5}+\sqrt{11})}{60}$

Q. If a and b rational numbers and $\frac{6}{3\sqrt{2}-2\sqrt{3}}=a\sqrt{2}-b\sqrt{3}$ then find the value of a and b.

Q.

Find the values of $a$ and $b$

if $\frac{5+2\sqrt{3}}{7+4\sqrt{3}}=a+b\sqrt{3}$

1. a= 11, b = -6

2. a = 9, b = -6

3. a= -11, b = 6

4. a= -9, b = 6

Q. If $\frac{3+\sqrt{5}}{2\sqrt{5}+3}=a+b\sqrt{5}$, then the values of irrational numbers $a$ and $b$ are
[3 Marks]

Q.

If $x=3+2\sqrt{2}$, then find the value of $(\sqrt{x}-\frac{1}{\sqrt{x}})$

1. $1$

2. $2$

3. $3$

4. $4$

Q. If $x=\frac{\sqrt{a+3b}+\sqrt{a-3b}}{\sqrt{a+3b}-\sqrt{a-3b}}$, prove that $3b{x}^2-2ax+3b=0$

Q.

$Rationalise:\frac{1}{\sqrt{3}-2\sqrt{11}}$

1. $\frac{\sqrt{3}+2\sqrt{11}}{43}$

2. $\frac{\sqrt{3}+2\sqrt{11}}{41}$

3. $\frac{\sqrt{3}+2\sqrt{11}}{-41}$

4. $\frac{\sqrt{3}+2\sqrt{11}}{-43}$

Q. If $\sqrt{2}=1.4142$...... then $\sqrt{\frac{\sqrt{2}-1}{\sqrt{2}+1}}$ is equal to

1. 2.4142
2. 0.4142
3. 0.1718
4. 5.8282

Q. If both a and b are rational numbers, find a and b respectively from the following.
$\frac{3-\sqrt{5}}{3+2\sqrt{5}}=a\sqrt{5}-b$

1. $\frac{19}{11},\frac{9}{11}$
2. $\frac{2}{11},-\frac{19}{11}$
3. $\frac{10}{11},\frac{21}{11}$
4. $\frac{9}{11},\frac{19}{11}$

Q. Simplify:

1. $5$
2. $2$
3. $1$
4. $0$

Q.

$\sqrt{3}\times 3\sqrt{3}$ is rational

1. True

2. False

Q.

Rationalising $\frac{\sqrt{2}+\sqrt{3}}{\sqrt{3}-\sqrt{2}}$ will give __________.

1. $7-2\sqrt{6}$

2. $5-2\sqrt{6}$

3. $5+2\sqrt{6}$

4. $7+2\sqrt{6}$

Q.

If $x=\frac{1}{\sqrt{3}+1}$ then the value of $x+\frac{1}{x}$ is

1. 

2. 

3. 

4. 

Q.

$\frac{(0.137{)}^3+(0.113{)}^3}{(0.137{)}^2-(0.137\left)\right(0.113)+(0.113{)}^2}=$

Q.

Simplify $\sqrt{\frac{12}{121}}$

1. $\frac{2\sqrt{3}}{11}$

2. $\frac{3\sqrt{2}}{11}$

3. $\frac{5\sqrt{2}}{11}$

4. None of these

Q. Simplify $\sqrt{3+2\sqrt{2}}$.

Q.

If 'a' and 'b' are rational numbers, then find the values of 'a' and 'b' given that $\frac{2+5\sqrt{3}}{2-\sqrt{3}}=a+b\sqrt{3}$

Q. Find the value of m .

Q.

The value of $\frac{(2.3{)}^3-0.027}{(2.3{)}^2+0.69+0.09}$is:

1. $2$

2. $2.327$

3. $3$

4. $2.273$