Surds
A surd is a square root which cannot be reduced to a rational number.
For example, 
is not a surd.
However 
is a surd.
If you use a calculator, you will see that 
and we will need to round the answer correct to a few decimal places. This makes it less accurate.
If it is left as 
, then the answer has not been rounded, which keeps it exact.
Here are some general rules when simplifying expressions involving surds.
- am x an = am + n
am | = am – n |
an | |
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- (am)n = amn
- (ab)n = anbn
| a |
| n | = | an |
b | bn | ||||
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- a0 = 1
Questions
Level-I
1. | (17)3.5 x (17)? = 178 | |||||||
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2. |
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3. | Given that 100.48 = x, 100.70 = y and xz = y2, then the value of z is close to: | |||||||
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4. | If 5a = 3125, then the value of 5(a – 3) is: | |||||||
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5. | If 3(x – y) = 27 and 3(x + y) = 243, then x is equal to: | |||||||
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.6. | (256)0.16 x (256)0.09 = ? | |||||||
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7. | The value of [(10)150 ÷ (10)146] | |||||||
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8. |
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9. | (25)7.5 x (5)2.5 ÷ (125)1.5 = 5? | |||||||||
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10. | (0.04)-1.5 = ? | |||||||
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Level-II
11. |
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12. |
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13. | If m and n are whole numbers such that mn = 121, the value of (m – 1)n + 1 is: | |||||||
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14. |
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15. If 5√5 * 53 ÷ 5-3/2 = 5a+2 , the value of a is:
A. 4
B. 5
C. 6
D. 8
A. 3
17. (ab)x−2=(ba)x−7. What is the value of x ?
A. 3
18. (0.04)-2.5 = ?
A. 125
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Answers
Level-I
Answer:1 Option D
Explanation:
Let (17)3.5 x (17)x = 178.
Then, (17)3.5 + x = 178.
3.5 + x = 8
x = (8 – 3.5)
x = 4.5
Answer:2 Option C
Explanation:
Given | a |
| x – 1 | = |
| b |
| x – 3 |
b | a |
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| a |
| x – 1 | = |
| a |
| -(x – 3) | = |
| a |
| (3 – x) |
b | b | b |
x – 1 = 3 – x
2x = 4
x = 2.
Answer:3 Option C
Explanation:
xz = y2 
10(0.48z) = 10(2 x 0.70) = 101.40
0.48z = 1.40
| 140 | = | 35 | = 2.9 (approx.) |
48 | 12 |
Answer:4 Option A
Explanation:
5a = 3125 
5a = 55
a = 5.
5(a – 3) = 5(5 – 3) = 52 = 25.
Answer:5 Option C
Explanation:
3x – y = 27 = 33 
x – y = 3 ….(i)
3x + y = 243 = 35 
x + y = 5 ….(ii)
On solving (i) and (ii), we get x = 4
Answer:6 Option A
Explanation:
(256)0.16 x (256)0.09 = (256)(0.16 + 0.09)
= (256)0.25
= (256)(25/100)
= (256)(1/4)
= (44)(1/4)
= 44(1/4)
= 41
= 4
Answer:7 Option B
Explanation:
(10)150 ÷ (10)146 = | 10150 |
10146 |
= 10150 – 146
= 104
= 10000.
Answer:8 Option B
Explanation:
Given Exp. = |
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= | xa | + | xb | + | xc |
(xa + xb + xc) | (xa + xb + xc) | (xa + xb + xc) |
= | (xa + xb + xc) |
(xa + xb + xc) |
= 1.
Answer:9 Option B
Explanation:
Let (25)7.5 x (5)2.5 ÷ (125)1.5 = 5x.
Then, | (52)7.5 x (5)2.5 | = 5x |
(53)1.5 |
| 5(2 x 7.5) x 52.5 | = 5x |
5(3 x 1.5) |
| 515 x 52.5 | = 5x |
54.5 |
5x = 5(15 + 2.5 – 4.5)
5x = 513
x = 13.
Answer:10 Option B
Explanation:
(0.04)-1.5 = |
| 4 |
| -1.5 |
100 |
= |
| 1 |
| -(3/2) |
25 |
= (25)(3/2)
= (52)(3/2)
= (5)2 x (3/2)
= 53
= 125.
Level-II
Answer:11 Option C
Explanation:
Given Expression |
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Answer:12 Option C
Explanation:
1 | + | 1 | = |
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1 + a(n – m) | 1 + a(m – n) |
= | am | + | an |
(am + an) | (am + an) |
= | (am + an) |
(am + an) |
= 1.
Answer:13 Option D
Explanation:
We know that 112 = 121.
Putting m = 11 and n = 2, we get:
(m – 1)n + 1 = (11 – 1)(2 + 1) = 103 = 1000.
Answer:14 Option B
Explanation:
Given Exp. |
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Answer:16
Explanation
am.an=am+n
(132)7 × (132)x = (132)11.5
=> 7 + x = 11.5
=> x = 11.5 – 7 = 4.5
Answer:17
Explanation:
an=1a−n
(ab)x−2=(ba)x−7⇒(ab)x−2=(ab)−(x−7)⇒x−2=−(x−7)⇒x−2=−x+7⇒x−2=−x+7⇒2x=9⇒x=92=4.5
Answer:18
Explanation:
a−n=1/an
(0.04)−2.5=(1/.04)2.5=(100/4)2.5=(25)2.5=(52)2.5=(52)(5/2)=55=3125
