Rationalizing Denominators using Rationalizing Factors

Q.

If ${(\sqrt{3}+i)}^{100}=2^{99}(a+ib)$, then $a^2+b^{2}is$

1. $\sqrt{2}$

2. $4$

3. $\sqrt{3}$

4. None of these

Q.

Prove that $\sqrt{2}+\sqrt{5}$ is irrational number

Q.

The product of $\sqrt{27}$ and $\sqrt{3}$ is ____________ which is a ___________ number $9$.

Q.

Multiplying $\frac{3}{\sqrt{17}-\sqrt{2}}$ by which fraction will produce an equivalent fraction with a rational denominator.

Q.

Choose and write the correct option. On rationalization of the denominator of the number $\frac{\sqrt{5}+2}{\sqrt{5}-2}$ we get:

1. ${\left(\sqrt{5}+2\right)}^2$

2. $\left(\sqrt{5}\right)2$

3. $3$

4. None of these

Q.

The complex number $\frac{\left(-\sqrt{3}+3i\right)\left(i-1\right)}{\left(3+\sqrt{3i}\right)\left(i\right)\left(\sqrt{3}+\sqrt{3i}\right)}$when represented in Argand diagram lies in which of the following.

1. In the first quadrant

2. In the second quadrant

3. On the Y-axis (imaginary axis)

4. On the X- axis (real axis)

Q.

Give an example of two irrational numbers to show that their sum is an irrational number.

Q. The rationalising factor for rationalization of the denominator of $\frac{21\sqrt{5}}{\sqrt{7}}$ is

1. $\sqrt{5}$
2. $\sqrt{7}$
3. $\sqrt{35}$
4. $\sqrt{21}$

Q. Write an equivalent expression by rationalizing the denominator $\frac{1}{6-\sqrt{10}}$

1. $\frac{6-\sqrt{10}}{26}$
   
2. $\frac{6+\sqrt{10}}{26}$
   
3. $\frac{26}{6+\sqrt{10}}$
   
4. $\frac{26}{6-\sqrt{10}}$
   

Q. Match the conjugate pairs.

1. $-\sqrt{7}+\sqrt{2}$
2. $\sqrt{8}+2\sqrt{2}$
3. $\sqrt{8}-2\sqrt{2}$
4. $\sqrt{7}+\sqrt{2}$

Q. Simplify the expression $\frac{\sqrt{3}-3\sqrt{6}}{\sqrt{3}+3\sqrt{6}}$ by rationalizing the denominator.

1. $\frac{6\sqrt{2}-19}{-17}$
2. $\frac{6\sqrt{2}-19}{17}$
3. $\frac{17\sqrt{2}-6}{17}$

Q. Match the numbers in the first column with their respective conjugate pairs.

1. $-\sqrt{7}$
2. $-\sqrt{9}$
3. $\sqrt{2}$

Q. Write an equivalent expression by rationalizing the denominator $\frac{1}{\sqrt{5}+\sqrt{3}}$

1. $\frac{\sqrt{5}-\sqrt{3}}{2}$
   
2. $\frac{\sqrt{5}+\sqrt{3}}{2}$
   
3. (a) and (b) above.
4. None of the above

Q. $\frac{9}{\sqrt{5}}\text{can be written as}\frac{9\sqrt{5}}{5}.$

1. True
2. False

Q. $\text{Rationalize the denominator of}\frac{1}{\sqrt{7}}$$\text{and express the equivalent number}.$

1. 
   $\frac{7}{\sqrt{7}}$
   
2. 
   $\frac{\sqrt{7}}{7}$
   
3. 
   $\left(a\right)$ and $\left(b\right)$ above.
4. 
   None of the above

Q. $\text{What is the outcome after the denominator of}\frac{1}{\sqrt{5}+2}$ $\text{has been rationalized?}$

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

Q. $\text{By rationalizing the denominator of}\frac{11}{\sqrt{7}},$$\text{we get}$

1. $\frac{7\sqrt{7}}{11}$
2. $\frac{11\sqrt{11}}{7}$
3. $\frac{7\sqrt{11}}{11}$
4. $\frac{11\sqrt{7}}{7}$

Q. Rationalize the denominator of the expression $\frac{4}{\sqrt{8}-\sqrt{2}}.$

1. $\frac{2(\sqrt{8}+\sqrt{2})}{3}$
2. $\frac{3(\sqrt{8}+\sqrt{2})}{2}$
3. $\frac{2}{3(\sqrt{8}+\sqrt{2})}$

Q. $\text{On rationalizing the denominator of}$$\frac{32}{\sqrt{8}},\text{we get}8\sqrt{2}$.

1. False
2. True

Q. For $\frac{3\sqrt{4}-\sqrt{6}}{3\sqrt{3}+4\sqrt{4}}$, choose the appropriate expression from the options below:

1. Rationalising factor is $3\sqrt{3}+4\sqrt{4}$
2. Rationalising factor is $3\sqrt{3}-4\sqrt{4}$
3. Rationalising factor is $3\sqrt{4}-\sqrt{6}$
4. Rationalising factor is $3\sqrt{4}+\sqrt{6}$

Q. In the following expression, find the values of $p$ and $q$

$\frac{\sqrt{2}+2}{\sqrt{2}-1}=p+q\sqrt{2}$

1. $p=0,q=1$
2. $p=1,q=0$
3. $p=1,q=1$
4. $p=4,q=3$

Q. Find the conjugate of $\sqrt{\frac{4}{3}}-\sqrt{\frac{3}{4}}$ and simplify it.

1. $\frac{7}{\sqrt{12}}$
2. $-\frac{7}{\sqrt{12}}$

Q. Simplify the expression: $\frac{1}{\frac{\sqrt{66}}{\sqrt{2}\times \sqrt{3}}+\sqrt[4]{4144}}$

1. $\sqrt{12}+\sqrt{11}$
2. $\sqrt{12}-\sqrt{11}$