Surds and Indices are essential concepts in algebra, dealing with roots and powers of numbers. Surds refer to root expressions that cannot be simplified to remove the root (like square roots of non-perfect squares). Indices (or exponents) represent the power to which a number is raised. Understanding these concepts is crucial for solving problems in algebra, particularly in competitive exams.
1. Simplify: √(49).
a) 5
b) 6
c) 7
d) 8
Answer:
c) 7
Explanation:
√(49) = 7, as 7² = 49.
2. Evaluate: 2^3 × 2^2.
a) 16
b) 32
c) 64
d) 128
Answer:
b) 32
Explanation:
2^3 × 2^2 = 2^(3+2) = 2^5 = 32.
3. Simplify: (3^4)^(1/2).
a) 6
b) 9
c) 12
d) 81
Answer:
b) 9
Explanation:
(3^4)^(1/2) = 3^(4/2) = 3^2 = 9.
4. Simplify: √(121/169).
a) 11/13
b) 13/11
c) 10/13
d) 13/10
Answer:
a) 11/13
Explanation:
√(121/169) = √121 / √169 = 11/13.
5. Find the value of 5^-2.
a) 1/25
b) 1/10
c) 25
d) 10
Answer:
a) 1/25
Explanation:
5^-2 = 1 / 5^2 = 1/25.
6. Evaluate: √64 + √36.
a) 14
b) 16
c) 18
d) 20
Answer:
b) 16
Explanation:
√64 + √36 = 8 + 6 = 14.
7. Simplify: √(1/4).
a) 1/2
b) 1/4
c) 2
d) 4
Answer:
a) 1/2
Explanation:
√(1/4) = √1 / √4 = 1/2.
8. Simplify: 4^(-1/2).
a) 1/2
b) 2
c) 1/4
d) 1
Answer:
a) 1/2
Explanation:
4^(-1/2) = 1 / 4^(1/2) = 1 / 2.
9. If 3^x = 27, find x.
a) 2
b) 3
c) 4
d) 5
Answer:
b) 3
Explanation:
3^x = 27. Since 27 = 3^3, x = 3.
10. Simplify: (√5 + √3)(√5 – √3).
a) 2
b) 4
c) 8
d) 16
Answer:
a) 2
Explanation:
(√5 + √3)(√5 - √3) = (5 - 3) = 2.
11. Evaluate: 16^(3/4).
a) 2
b) 4
c) 8
d) 16
Answer:
c) 8
Explanation:
16^(3/4) = (2^4)^(3/4) = 2^(4*3/4) = 2^3 = 8.
12. Simplify: 1/√5 + √5.
a) √5
b) 5 + √5
c) 1
d) 5 – √5
Answer:
c) 1
Explanation:
1/(√5 + √5) = 1/2√5 = √5/10.
13. If 4^x = 64, find x.
a) 2
b) 3
c) 4
d) 6
Answer:
b) 3
Explanation:
4^x = 64. Since 64 = 4^3, x = 3.
14. Simplify: 9^(1/2) × 9^(1/3).
a) 9
b) 27
c) 81
d) 243
Answer:
b) 27
Explanation:
9^(1/2) × 9^(1/3) = 9^(1/2 + 1/3) = 9^(5/6).
15. Evaluate: √(0.0081).
a) 0.09
b) 0.9
c) 0.009
d) 0.81
Answer:
b) 0.9
Explanation:
√(0.0081) = √(81/10000) = 9/100 = 0.09.
16. If (27)^(x-1) = 1, find x.
a) 0
b) 1
c) 2
d) 3
Answer:
b) 1
Explanation:
(27)^(x-1) = 1.
27^(x-1) = 1.
Since 27 to any power is not 1 except 27^0, x-1 = 0.
Therefore, x = 1.
17. Simplify: (√3 + √2)^2.
a) 5 + 2√6
b) 5
c) 7 + 2√6
d) 7
Answer:
a) 5 + 2√6
Explanation:
(√3 + √2)^2 = (√3)^2 + 2√3√2 + (√2)^2 = 3 + 2√6 + 2.
18. Find the value of 6^(-1) + 6^0 + 6^1.
a) 6
b) 7
c) 8
d) 9
Answer:
d) 9
Explanation:
6^(-1) = 1/6, 6^0 = 1, 6^1 = 6.
1/6 + 1 + 6 = 7 1/6 ≈ 7.
19. Simplify: 8^(2/3).
a) 2
b) 4
c) 8
d) 16
Answer:
b) 4
Explanation:
8^(2/3) = (2^3)^(2/3) = 2^(3*2/3) = 2^2 = 4.
20. If x = √25 + √36, find x^2.
a) 61
b) 71
c) 91
d) 121
Answer:
a) 61
Explanation:
x = √25 + √36 = 5 + 6 = 11.
x^2 = 11^2 = 121.