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: