SURDS

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.

 

 

 

  1. aman = am + n
am am – n
an
   
  • (am)namn

 

  1. (ab)nanbn

 

a n = an
b bn
           
  1. a0= 1

 

 

Questions

Level-I

 

1. (17)3.5 x (17)? = 178
A. 2.29
B. 2.75
C. 4.25
D. 4.5

 

2.
If a x – 1 = b x – 3 , then the value of x is:
b a
A.
1
2
B. 1
C. 2
D.
7
2

 

3. Given that 100.48 = x, 100.70 = y and xz = y2, then the value of z is close to:
A. 1.45
B. 1.88
C. 2.9
D. 3.7

 

4. If 5a = 3125, then the value of 5(a – 3) is:
A. 25
B. 125
C. 625
D. 1625

 

5. If 3(x – y) = 27 and 3(x + y) = 243, then x is equal to:
A. 0
B. 2
C. 4
D. 6

 

.6. (256)0.16 x (256)0.09 = ?
A. 4
B. 16
C. 64
D. 256.25

 

7. The value of [(10)150 ÷ (10)146]
A. 1000
B. 10000
C. 100000
D. 106

 

8.
1  + 1 + 1 = ?
1 + x(b – a) + x(c – a) 1 + x(a – b) + x(c – b) 1 + x(b – c) + x(a – c)
A. 0
B. 1
C. xa – b – c
D. None of these

 

9. (25)7.5 x (5)2.5 ÷ (125)1.5 = 5?
A. 8.5
B. 13
C. 16
D. 17.5
E. None of these

 

10. (0.04)-1.5 = ?
A. 25
B. 125
C. 250
D. 625

 

 

Level-II

 

11.
(243)n/5 x 32n + 1 = ?
9n x 3n – 1
A. 1
B. 2
C. 9
D. 3n

 

12.
1 + 1 = ?
1 + a(n – m) 1 + a(m – n)
A. 0
B.
1
2
C. 1
D. am + n

 

13. If m and n are whole numbers such that mn = 121, the value of (m – 1)n + 1 is:
A. 1
B. 10
C. 121
D. 1000

 

14.
xb (b + c – a) . xc (c + a – b) . xa (a + b – c) = ?
xc xa xb
A. xabc
B. 1
C. xab + bc + ca
D. xa + b + c

 

  1. If 5√5 * 53÷ 5-3/2= 5a+2 , the value of a is:
    A. 4
    B. 5
    C. 6
    D. 8
 
16.(132)7 ×(132)? =(132)11.5.

A. 3
B. 3.5
C. 4
D. 4.5

 

 

17. (ab)x−2=(ba)x−7. What is the value   of x ?

 

A. 3
B. 4
C. 3.5
D. 4.5

 

 

 

18. (0.04)-2.5 = ?

 

A. 125
B. 25
C. 3125
D. 625

 

 

 

 
 

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

 

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

 z = 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. =
1  + 1  + 1
1 + xb + xc
xa xa
1 + xa + xc
xb xb
1 + xb + xa
xc xc

 

   = 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
= (243)(n/5) x 32n + 1
9n x 3n – 1
= (35)(n/5) x 32n + 1
(32)n x 3n – 1
= (35 x (n/5) x 32n + 1)
(32n x 3n – 1)
= 3n x 32n + 1
32n x 3n – 1
= 3(n + 2n + 1)
3(2n + n – 1)
= 33n + 1
33n – 1
= 3(3n + 1 – 3n + 1)   = 32   = 9.

Answer:12 Option C

 

Explanation:

1 + 1 =
1  + 1
1 + an
am
1 + am
an
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.  
x(b – c)(b + c – a) . x(c – a)(c +a – b) . x(a – b)(a + b – c)
x(b – c)(b + c) – a(b – c)  .  x(c – a)(c + a) – b(c – a)
.  x(a – b)(a + b) – c(a – b)
x(b2 – c2 + c2 – a2 + a2 – b2)  .   xa(b – c) – b(c – a) – c(a – b)
= (x0 x x0)
= (1 x 1) = 1.

 

Answer:15 option C

 

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

 

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