Rationalisation of Denominator

Q.

If a and b are two rational numbers prove that a+b , a-b , ab are rational numbers. If b is not equal to 0 show that a/b is also a rational number

Q.

Find the value of $\frac{1}{3-\sqrt{8}}-\frac{1}{\sqrt{8}-\sqrt{7}}+\frac{1}{\sqrt{7}-\sqrt{6}}-\frac{1}{\sqrt{6}-\sqrt{5}}+\frac{1}{\sqrt{5}-2}$ .

1. 1

2. 5

3. 6

4. 4

Q. Let x= p/q be a rational number, such that the prime factorisation of q is not of the form $2^m{5}^n$, where n, m are non-negative integers. Then, x has a decimal expansion which is ___.

1. Non-terminating repeating (recurring)
2. Terminating non-repeating (non-recurring)
3. Real Integer
4. None of these

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.

Question 14
After rationalizing the denominator of $\frac{7}{3\sqrt{3}-2\sqrt{2}}$, we get the denominator as

A) 13
B) 19
C) 5
D) 35

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

Q.

Question 5 (iv)
Rationalize the denominators of the following:

$\frac{1}{(\sqrt{7}-2)}$

Q.

Question 5 (i)
Rationalize the denominator of $\frac{1}{\sqrt{7}}$ .

Q.

Question 12
The number obtained on rationalising the denominator of $\frac{1}{\sqrt{7}-2}$ is

A) $\frac{\sqrt{7}+2}{3}$
B) $\frac{\sqrt{7}-2}{3}$
C) $\frac{\sqrt{7}+2}{5}$
D) $\frac{\sqrt{7}+2}{45}$

Q.

Find the values of $a$ and $b$

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

1. a= -9, b = 6

2. a= 11, b = -6

3. a = 9, b = -6

4. a= -11, b = 6

Q. Product of a binomial surd and its conjugate is always ________ .

1. an integer
2. a rational number
3. an irrational number
4. a surd

Q.

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

1. 11
2. 9
3. 6
4. -6

Q.

Question 5 (ii)
Rationalize the denominator of :
$\frac{1}{(\sqrt{7}-\sqrt{6})}$ .

Q. Question 11
Find the values of a and b in each of the following
$\left(i\right)\frac{5+2\sqrt{3}}{7+4\sqrt{3}}=a-6\sqrt{3}$
$\left(ii\right)\frac{3-\sqrt{5}}{3+2\sqrt{5}}=a\sqrt{5}-\frac{19}{11}$
$\left(iii\right)\frac{\sqrt{2}+\sqrt{3}}{3\sqrt{2}-2\sqrt{3}}=2-b\sqrt{6}$
$\left(iv\right)\frac{7+\sqrt{5}}{7-\sqrt{5}}-\frac{7-\sqrt{5}}{7+\sqrt{5}}=a+\frac{7}{11}\sqrt{5}b$

Q.

Question 5 (iii)
Rationalize the denominators of the following:

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

Q. If $\sqrt{2}=1.414,\sqrt{3}=1.732,\sqrt{5}=2.236$ and $\sqrt{6}=2.449,$ find the value of $\frac{2+\sqrt{3}}{2-\sqrt{3}}+\frac{2-\sqrt{3}}{2+\sqrt{3}}+\frac{\sqrt{3}-1}{\sqrt{3}+1}$ .

Q.

Rationalize the denominator of :
$\frac{1}{(\sqrt{7}-\sqrt{6})}$ .

Q. Which of the following is equivalent to $\frac{1}{\sqrt{6}}+\frac{1}{\sqrt{5}}$ ?

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

Q. Rationalise the denominator and simplify:

Q. Prove that root 9 is rational.

Q. If $\sqrt{2}=1.414,\sqrt{3}=1.732,\sqrt{5}=2.236$ and $\sqrt{6}=2.449,$ find the value of $\frac{2+\sqrt{3}}{2-\sqrt{3}}+\frac{2-\sqrt{3}}{2+\sqrt{3}}+\frac{\sqrt{3}-1}{\sqrt{3}+1}$ .

1. 14.268
2. 18.428
3. 16.629
4. 14.662

Q.

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

1. $5-2\sqrt{6}$

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

3. $7+2\sqrt{6}$

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

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.

$\frac{2\sqrt{3}}{\sqrt{7}}=$ ___

1. $1$

2. $\frac{2\sqrt{21}}{7}$

3. $2\sqrt{21}$

4. $\frac{2\sqrt{7}}{\sqrt{3}}$

Q.

$\frac{4}{1-\sqrt{2}-\sqrt{3}}=$ ____________.

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

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

3. 4

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

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.

$\text{If}y=\frac{\sqrt{5}+2}{\sqrt{5}-2}$, then $y^2=$ _________.

1. $161+72\sqrt{5}$

2. $161$

3. $161-72\sqrt{5}$

4. $144\sqrt{5}$

Q.

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

1. $18+8\sqrt{5}$

2. $18-8\sqrt{5}$

3. + 18

4. -18

Q.

$\text{If}m=\frac{1}{3-2\sqrt{2}}\text{then,}{m}^2=$ _______.

1. $34$

2. $17-12\sqrt{2}$

3. $24\sqrt{2}$

4. $17+12\sqrt{2}$