2. Chapter 2. Exponents and Radicals NIOS Class 10 Mathematics Textbook Solutions
Chapter 2. Exponents and Radicals NIOS Class 10 Mathematics Textbook Solutions for Exam Preparation
Chapter 2. Exponents and Radicals
CHECK YOUR PROGRESS 2.1 EXPONENTS AND RADICALS
1. Write the following in exponential form:
(i) 
(ii) 
(iii) 
Solution: (i) 
(ii) 
(iii) 
2. Write the base and exponent in each of the following:
(i) 
(ii) 
(iii) 
Solution: (i) 
The base = – 3 and the exponent = 5
(ii) 
The base = 7 and the exponent = 4
(iii) 
The base 
and the exponent = 8
3. Evaluate each of the following :
(i) 
(ii) 
(iii) 
Solution: (i) We have, 
(ii) We have, 
(iii) We have, 
4. Simplify the following: (i) 
(ii) 
Solution: (i)
(ii) We have,
5. Find the reciprocal of each of the following: (i) 
(ii) 
(iii) 
Solution: (i) 
The reciprocal of 
(ii)
The reciprocal of 
(iii)
The reciprocal of 
CHECK YOUR PROGRESS 2.2 EXPONENTS AND RADICALS
1.Express each of the following as a product of powers of primes, i.e, in exponential form:
(i) 429 (ii) 648 (iii) 1512
Solution: (i) We have, 429 = 3 × 11 × 13
(ii) We have, 
(iii) We have, 
2. Express each of the following in exponential form:
(i) 729 (ii) 512 (iii) 2592 (iv) 
(v) 
Solution: (i) We have, 
(ii) We have, 
(iii) We have, 
(iv) We have, 
(v) We have, 
CHECK YOUR PROGRESS 2.3 EXPONENTS AND RADICALS
1. Simplify and express the result in exponential form:
(i) 
(ii) 
(iii) 
Solution: (i) We have, 
(ii) We have, 
(iii) We have, 
2. Simplify and express the result in exponential form:
(i) 
(ii) 
(iii) 
Solution: (i) We have, 
(ii) We have, 
(iii) We have, 
[ Note: 
]
3. Simplify and express the result in exponential form:
(i) 
(ii) 
(iii) 
(iv) 
(v) 
Solution: (i) We have, 
(ii) We have, 
(iii) We have, 
(iv) We have, 
(v) We have, 
[ Note: 
; 
; 
]
4. Which of the following statements are true?
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
(vii) 
Solution: (i) 
→ True .
(ii) 
→ True.
(iii) 
→ False.
(iv) 
→ False.
(v) 
→ False.
(vi) 
→ False .
(vii) 
→ True.
CHECK YOUR PROGRESS 2.4 EXPONENTS AND RADICALS
1. Express 
as a rational number of the form 
:
Solution: We have, 
2. Express as a power of rational number with positive exponent:
(i) 
(ii) 
(iii) 
[ Note : 
]
Solution: (i) We have, 
(ii) We have, 
(iii) We have, 
3. Express as a power of a rational number with negative index:
(i) 
(ii) 
(iii) 
[ Note: 
,
, 
]
Solution: (i) We have, 
(ii) We have, 
(iii) We have, 
4. Simplify: (i) 
(ii) 
(iii) 
Solution: (i) We have, 
(ii) We have, 
(iii) We have, 
5. Which of the following statements are true?
(i) 
(ii) 
(iii) 
(iv) 
(v) 
Solution: (i) 
→ False .
(ii) → True
(iii) → True
(iv) → true
(v) 
→ False
CHECK YOUR PROGRESS 2.5 EXPONENTS AND RADICALS
1. Simplify each of the following: (i) 
(ii) 
Solution: (i) We have, 
(ii) We have, 
2. Simplify each of the following:
(i) 
(ii) 
(iii) 
Solution: (i) We have, 
(ii) We have, 
(iii) We have, 
CHECK YOUR PROGRESS 2.6 EXPONENTS AND RADICALS
1. For each of the following, write index and the radicand:
(i) 
(ii) 
(iii) 
[ Note: 
]
Solution: (i)
The index is 4 and radicand is 64.
(ii)
The index is 6 and radicand is 343 .
(iii)
The index is 1 and radicand is 119.
2. State which of the following are surds:
(i) 
(ii) 
(iii) 
(iv) 
(v) 
Solution: (i) We have, 
It is not a surd.
(ii) We have, 
It is not a surd.
(iii) We have, 
It is a surd.
(iv) We have, 
It is not a surd.
(v) We have, 
It is a surd.
3. Identify pure and mixed surds out of the following:
(i) 
(ii) 
(iii) 
(iv) 
Solution: (i) 
is a mixed surd.
(ii) 
is a mixed surd.
(iii) 
is a mixed surd.
(iv) 
is a pure surd.
CHECK YOUR PROGRESS 2.7 EXPONENTS AND RADICALS
1. State which of the following are pairs of similar surds:
(i) 
(ii) 
(iii) 
Solution: (i) 
and
Therefore, 
are pairs similar surds .
(ii)
Therefore, 
are not pairs similar surds .
(iii)
Therefore, 
are pairs similar surds .
2. Express as a pure surd:
(i) 
(ii) 
(iii) 
Solution: (i) We have, 
(ii) We have, 
(iii) We have, 
3. Express as a mixed surd in the simplest form:
(i) 
(ii) 
(iii) 
Solution: (i) we have, 
(ii) We have, 
(iii) we have, 
CHECK YOUR PROGRESS 2.8 EXPONENTS AND RADICALS
Simplify each of the following:
1. 
2. 
3. 
4. 
5. 
6. 
7. 
Solution: 1. We have, 
2. We have, 
3. We have, 
4. We have, 
5. We have, 
6. We have,
7. We have, 
CHECK YOUR PROGRESS 2.9 EXPONENTS AND RADICALS
1. Multipliy 
.
Solution: We have, 
2. Multipliy 
.
Solution: We have, 
3. Divide 
.
Solution: We have, 
4. Divide 
.
Solution: We have, 
5. Which is greater 
?
Solution: We have, 
and 
6. Which in smaller: 
?
Solution: We have, 
and 
7. Arrange in ascending order:
Solution: We have,
; 
and 
8. Arrange in descending order: 
Solution: We have, 
and 
CHECK YOUR PROGRESS 2.10
1. Find the rationalising factor of each of the following:
(i) 
(ii) 
(iii) 
Solution: (i) We have, 
Therefore, the rationalising factor is 
.
(ii) We have, 
Therefore, the rationalising factor is 
.
(iii) We have, 
2. Simplify by rationalising the denominator of each of the following:
(i) 
(ii) 
(iii) 
(iv) 
Solution: (i) We have, 
(ii) We have, 
(iii) We have, 
(iv) We have, 
3. Simplify: 
Solution: We have, 
4. Rationalise the denominator of 
Solution: We have, 
5. If 
. Find 
Solution: We have,
6. If 
, find x and y .
Solution: We have,
and 
TERMINAL EXERCISE
1. Express the following in exponential form:
(i) 
(ii) 
Solution: (i) We have, 
(ii) We have, 
2. Simplify the following:
(i) 
(ii) 
Solution: (i) We have, 
(ii) We have, 
3. Simplify and express the result in exponential form:
(i) 
(ii) 
(iii) 
Solution: (i) We have, 
(ii) We have, 
(iii) We have, 
4. Simplify each of the following:
(i) 
(ii) 
Solution: (i) We have, 
(ii) We have, 
5. Simplify the following:
(i) 
(ii) 
(iii) 
Solution: (i) We have, 
(ii) We have, 
(iii) We have, 
6. Find x so that 
Solution: We have, 
7. Find x so that 
Solution: We have,
8. Express as a product of primes and write the answers of each of the following in exponential form:
(i) 6480000 (ii) 172872 (iii) 11863800
Solution: (i) We have, 
(ii) We have, 
(iii) We have, 
9. The star sirus is about 
km from the earth. Assuming that the light travels at 
km per second, find how long light from sirus takes to reach earth.
Solution: Here, the distance 
km and the speed 
km/s
We have, 
10. State which of the following are surds:
(i) 
(ii) 
(iii) 
(iv) 
Solution: (i) 
is not a surd.
(ii) 
is a surd.
(iii) 
is a surd.
(iv)
is a mixed surd.
11. Express as a pure surd:
(i) 
(ii) 
(iii) 
Solution: (i) 
(ii) 
(iii) 
12. Express as a mixed surd in simplest form:
(i) 
(ii) 
(iii) 
Solution: (i) 
(ii) 
(iii)
13. Which of the following are pairs of similar surds?
(i) 
(ii) 
(iii)
Solution: (i)
and 
Therefore, 
are similar surds .
(ii)
and 
Therefore, 
are similar surds .
(iii)
and 
Therefore, 
are not similar surds .
14.Simplify each of the following:
(i) 
(ii)
(iii) 
Solution: (i) We have, 
(ii) We have, 
(iii) We have,
15. Which is greater?
(i) 
(ii) 
Solution: (i) 
LCM of 2 and 3 is 6.
(ii) 
LCM of 3 and 4 is 12 .
16. Arrange in descending order:
(i) 
(ii) 
Solution: (i)
LCM of 2, 3, and 4 is 12 .
(ii)
LCM of 2, 2 and 3 is 6.
17. Arrange in ascending order: 
Solution: We have, 
and 
18. Simplify by rationalising the denominator:
(i) 
(ii) 
(iii) 
Solution: (i) We have, 
(ii) We have, 
(iii) We have, 
19. Simplify each of the following by rationalising the denominator:
(i) 
(ii) 
Solution: (i) We have, 
(ii) We have, 
20. If 
, find the values of a and b, where a and b are rational numbers.
Solution: We have,
and 
21. If 
, find the value of 
.
Solution: We have,
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