Question 1: State with reason which of the following are surds.

Answer:

. Therefore it is surd.

. This is a surd.

: We observe that is an irrational number. But, is not a rational number. Hence is not a surd.

. cannot be expresses as a rational number under a root sign. Therefore is not a surd.

. Therefore it is a surd.

Question 2: Simplify the following:

Answer:

Question 3: Express the following as pure surds:

Answer:

Question 4: Express each of the following as a mixed surd in simplest form:

Answer:

Question 5: Convert:

into a surd of order

into a surd of order

and into surds of the same but smallest order

and into surds of the same but smallest order

into a surd of order

Answer:

into a surd of order

into a surd of order

and into surds of the same but smallest order

LCM of and is

and into surds of the same but smallest order

LCM of and is

into a surd of order

Question 6: Which is greater?

Answer:

LCM of and is

LCM of and is

LCM of and is

First simplify each of the given terms

For both the terms, the numerator is the same which is 4. Therefore whichever term has a higher denominator, would be the smaller term. Let’s compare the two denominators.

First simplify each of the given terms

For both the terms, the numerator is the same which is 5. Therefore whichever term has a higher denominator, would be the smaller term. Let’s compare the two denominators.

Question 7: Arrange in Ascending Order:

Answer:

LCM of

Now convert all the above terms to order of 12

Now comparing the number under the root sign as they are all of the same order.

or

LCM of

Now convert all the above terms to order of 12

Now comparing the number under the root sign as they are all of the same order.

or

Question 8: Arrange in Descending Order:

Answer:

Convert into simple surds

Since the order of all the terms is the same, just compare the terms inside the square root. Hence, the descending order is

LCM of

Now convert all the above terms to order of 24

Now comparing the number under the root sign as they are all of the same order.

or

Question 9: Simplify:

Answer:

Question 10: Multiply:

Answer:

LCM of

Question 11: Divide:

Answer:

Question 12: Find the rationalising factors of the following:

Answer:

.

We know that the rationalizing factor of monomial is . Therefore the monomial the rationalizing factor should be

.

We know that the rationalizing factor of monomial is . Therefore the monomial the rationalizing factor should be

We find that

Rationalizing factor of is .

Hence is the rationalizing factor of

We have

The conjugate of is

Therefore the rationalizing factor of is .

We know that the rationalizing factor of monomial is . Therefore the monomial the rationalizing factor should be

Question 13: Rationalize the denominator and simplify:

Answer:

Question 14: Simplify:

Answer:

Question 15: Determine rational numbers and

Answer:

and

and

Let’s first simplify

Now comparing,

Answer:

Answer:

Answer:

Therefore

Hence

Answer:

Therefore

Hence

Answer:

Question 21: If , find the value of

Answer:

Question 22: Given and , find:

Answer:

Question 23: Rationalize and simplify:

Answer: