# 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

Level-I

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

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.

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

Explanation:

5a = 3125        5a = 55

a = 5.

5(a – 3) = 5(5 – 3) = 52 = 25.

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

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

Explanation:

 (10)150 ÷ (10)146 = 10150 10146

= 10150 – 146

= 104

= 10000.

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.

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.

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

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.

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.

Explanation:

We know that 112 = 121.

Putting m = 11 and n = 2, we get:

(m – 1)n + 1 = (11 – 1)(2 + 1) = 103 = 1000.

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)  .   x–a(b – c) – b(c – a) – c(a – b)
 = (x0 x x0)
 = (1 x 1) = 1.

Explanation

am.an=am+n

(132)7 × (132)x = (132)11.5

=> 7 + x = 11.5

=> x = 11.5 – 7 = 4.5

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

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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