Rationalisation

Q.

If $x=2+\sqrt{3}$, find the value of $x^3+\frac{1}{x^3}$.

Q.

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

Q.

If $x=2+\sqrt{3}$, find the value of $x+\frac{1}{x}$.

Q.

If $x=3+\sqrt{8},$ find the value of $x^2+\frac{1}{x^2}$.

Q.

If x = 9-4$\sqrt{5}$, find the value of $x^2+\frac{1}{x^2}$

Q.

The value of $\sqrt{3-2\sqrt{2}}$ is

1. $\sqrt{2}-1$

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

3. $\sqrt{2}+1$

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

Q.

If $x=\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$ and $y=\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$, then x + y + xy =

1. 9

2. 5

3. 17

4. 7

Q.

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

1. 2.4142

2. 0.1718

3. 5.8282

4. 0.4142

Q. Question 13
Rationalise the denominator in each of the following and hence evaluate by taking $\sqrt{2}=1.414,\sqrt{3}=1.732$ and $\sqrt{5}=2.236$ upto three places of decimal.
$\left(i\right)\frac{4}{\sqrt{3}}\left(ii\right)\frac{6}{\sqrt{6}}\left(iii\right)\frac{\sqrt{10}-\sqrt{5}}{2}\left(iv\right)\frac{\sqrt{2}}{2+\sqrt{2}}\left(v\right)\frac{1}{\sqrt{3}+\sqrt{2}}$

Q.

Find the value to three places of decimals of each of the following. It is given that $\sqrt{2}=1.414,\sqrt{3}=1.732,\sqrt{5}=2.236$ and $\sqrt{10}=3.162$.

(i) $\frac{2}{\sqrt{3}}$

(ii) $\frac{3}{\sqrt{10}}$

(iii) $\frac{\sqrt{5}+1}{\sqrt{2}}$

(iv) $\frac{\sqrt{10}+\sqrt{15}}{\sqrt{2}}$

(v) $\frac{2+\sqrt{3}}{3}$

(vi) $\frac{\sqrt{2}-1}{\sqrt{5}}$

Q.

If $\frac{5+2\sqrt{3}}{7+4\sqrt{3}}=a+b\sqrt{3}$ , then the value of $a$ and $b$ is

Q.

Find the value of $\frac{6}{\sqrt{5}-\sqrt{3}}$, it being given that $\sqrt{3}=1.732and\sqrt{5}=2.236$.

Q.

The conjugate of a complex number is $\frac{1}{(i-1)}$, then that complex number is

1. $\frac{1}{(i-1)}$

2. $\frac{-1}{(i-1)}$

3. $\frac{1}{(i+1)}$

4. $\frac{-1}{(i+1)}$

Q. Simplify : $\frac{4+\sqrt{5}}{4-\sqrt{5}}+\frac{4-\sqrt{5}}{4+\sqrt{5}}.$

1. 
   $\frac{40}{11}$
2. 
   $\frac{42}{11}$
3. 
   $\frac{43}{11}$
4. 
   $\frac{39}{11}$

Q. Question 9
Simplify the following
$\left(i\right)\sqrt{45}-3\sqrt{20}+4\sqrt{5}\left(ii\right)\frac{\sqrt{24}}{8}+\frac{\sqrt{54}}{9}\left(iii\right)4\sqrt{12}\times 7\sqrt{6}\left(iv\right)4\sqrt{28}+3\sqrt{7}+\sqrt[3]{37}\left(v\right)3\sqrt{3}+2\sqrt{27}+\frac{7}{\sqrt{3}}\left(vi\right)(\sqrt{3}-\sqrt{2}{)}^2(vii)\sqrt[4]{481}-8\sqrt[3]{3216}+15\sqrt[5]{532}+\sqrt{225}(viii)\frac{3}{\sqrt{8}}+\frac{1}{\sqrt{2}}(ix)\frac{2\sqrt{3}}{3}-\frac{\sqrt{3}}{6}$

Q.

If $\sqrt{2}=1.41$ then $\frac{1}{\sqrt{2}}$=?

(a) 0.075 (b) 0.75 (c) 0.705 (d) 7.05

Q.

Express each one of the following with rational denominator :

(i) $\frac{1}{3+\sqrt{2}}$

(ii) $\frac{1}{\sqrt{6}-\sqrt{5}}$

(iii) $\frac{16}{\sqrt{41}-5}$

(iv) $\frac{30}{5\sqrt{3}-3\sqrt{5}}$

(v) $\frac{1}{2\sqrt{5}+\sqrt{3}}$

(vi) $\frac{3\sqrt{3}+1}{2\sqrt{2}-\sqrt{3}}$

(vii) $\frac{6-4\sqrt{2}}{6+4\sqrt{2}}$

(viii) $\frac{3\sqrt{2}+1}{2\sqrt{5}-3}$

(ix) $\frac{b^2}{\sqrt{a^2+b^2+a}}$

Q.

If $a=\sqrt{2}+1$, then find the value of $a-\frac{1}{a}$.

Q.

If $\sqrt{2}=1.414,$ then the value of $\sqrt{6}-\sqrt{3}$ upto three places of decimal is ______ .

1. 0.235

2. 0.717

3. 1.414

4. 0.471

Q.

If $\frac{\sqrt{3}-1}{\sqrt{3}+1}=x+y\sqrt{3}$, find the value of x and y.

Q.

The rationalising factor for the number $3-7\sqrt{2}$ is

1. $3+2\sqrt{7}$

2. $-3-2\sqrt{7}$

3. $3+7\sqrt{2}$

4. $-3+7\sqrt{2}$

Q.

The simplest rationalising factor of $\sqrt{3}+\sqrt{5}$ is

1. $\sqrt{3}-5$

2. $3-\sqrt{5}$

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

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

Q. Rationalise the denominator and simplify:

\(\dfrac{4\sqrt{3} +5\sqrt{2}}{\sqrt{48} + \sqrt{18}}\)

Q.

The simplest rationalising factor of $2\sqrt{5}-\sqrt{3}$ is

1. $2\sqrt{5}+3$

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

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

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

Q.

Find the values of each of the following correct to three places of decimals, it being given that $\sqrt{2}=1.4142,\sqrt{3}=1.732,\sqrt{5}=2.2360,\sqrt{6}=2.4492and\sqrt{10}=3.162$.

(i) $\frac{3-\sqrt{5}}{3+2\sqrt{5}}$

(ii) $\frac{1+\sqrt{2}}{3-2\sqrt{2}}$

Q.

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

1. $-2\sqrt{3}$

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

3. $2\sqrt{3}$

4. $-\sqrt{3}$

Q.

Write the reciprocal of $5+\sqrt{2}$.

Q.

In each of the following determine rational numbers a and b :

(i) $\frac{\sqrt{3}-1}{\sqrt{3}+1}=a-b\sqrt{3}$

(ii) $\frac{4+\sqrt{2}}{2+\sqrt{2}}=a-\sqrt{b}$

(iii) $\frac{3+\sqrt{2}}{3-\sqrt{2}}=a+b\sqrt{2}$

(iv) $\frac{5+3\sqrt{3}}{7+4\sqrt{3}}=a+b\sqrt{3}$

(v) $\frac{\sqrt{11}-\sqrt{7}}{\sqrt{11}+\sqrt{7}}=a-b\sqrt{77}$

(vi) $\frac{4+3\sqrt{5}}{4-3\sqrt{5}}=a+b\sqrt{5}$

Q.

If $x=\sqrt{6}+\sqrt{5},$ then $x^2+\frac{1}{x^2}-2=$

1. $2\sqrt{5}$

2. $2\sqrt{6}$

3. 24

4. 20

Q.

If $x=\sqrt[3]{32+\sqrt{3}},then{x}^3+\frac{1}{x^3}=$

1. 4

2. 8

3. 9

4. 2