Question 1

Solve the following: ? =?103823

Accepted Answer

? = ?103823 ? = ?(47) 3 Apply the Law of Algebra ? = [(47)3]1/3 ? = 473 x 1/3 = 47 ? = 47

Question 2

If (1/5 )3a = 0.008 Find the value of ( 0.25)a

Accepted Answer

(1/5 )3a = 0.008 = 8/1000 = 1/125 = (1/5 )3 Apply the Algebra Law and Solve the equation. ⇒ 3a = 3 ∴ a = 1 ∴ ( 0.25 )a = ( 0.25 )1 = 0.25

Question 3

If (p/q)n - 1 = (q/p)n - 3, then the value of n is

Accepted Answer

Given equation is below (p/q)n - 1 = (q/p)n - 3 Apply the law of Fractional Exponents and Laws of Exponents a - m = 1/a m or am = 1/a - m (p/q)n - 1 = (q) - (n - 3) x 1/ (p) - (n - 3) (p/q)n - 1 = (p) 3 - n x 1/ (q) 3 - n (p/q)n - 1 = (p/q) 3 - n if pX = pY then X will be equal to Y. means X = Y; ⇒ n - 1 = 3 - n ⇒ 2n = 4 ⇒ n = 2

Question 4

If a = √3/2, then √1 + a + √1 - a =?

Accepted Answer

Let us assume P = √1 + a + √1 - a Square on both side, P 2 = ( √1 + a + √1 - a ) 2 Use the formula ( a + b ) 2 = a 2 + b 2 + 2ab we will get ⇒ P 2 = (1 + a) + (1- a) + 2 √1 + a x √1 - a ⇒ P 2 = (1 + a) + (1- a) + 2 √(1 + a ) x ( 1 - a ) ⇒ P 2 = 2 + 2 √1 - a2 ⇒ P 2 = 2 (1 + √1 - a2) Now put the value of a which is given in question; ⇒ P 2 = 2 (1 + √1 - ( √3/2 )2) ⇒ P 2 = 2(1 + √1 - 3/4 ) ⇒ P 2 = 2(1 + √1/4 ) ⇒ P 2 = 2(1 + 1/2) = 2 x 3/2 = 3 ⇒ P= √3 ∴ (√1 + a + √1 - a) = √3

Question 5

1/1 + xb - a + xc - a + 1/1 + xa-b + xc - b + 1/1 + xb - c+ xa - c =?

Accepted Answer

? = 1/ ( 1 + xb - a + xc - a ) + 1/ ( 1 + xa - b + xc - b ) + 1 / ( 1 + xb - c + xa - c ) Apply the law of algebra and solve the given equation. ⇒ ? =1/ ( 1 + xb/xa + xc/xa ) + 1/ ( 1+ xa/xb + xc/xb ) + 1/ ( 1 + xb/xc + xa/xc ) ⇒ ? = xa/ ( xa+xb+xc ) + xb/ ( xb + xa+ xc ) + xc/ ( xc+xb+xa ) ⇒ ? = ( xa + xb + xc ) / ( xa + xb + xc ) = 1

Question 6

if a = 2 + √3, then what is the value of (a2 + a -2)?

Accepted Answer

Given that, a = 2 + √ 3 Then, 1/a = 2 - √3 [ by conjugate property ] Conjugate law is below, if x = p + √ q then 1/x = p - √ q Apply the law of algebra in above equation. Now, we have, a 2 + a -2 = ( a + 1/a ) 2 - 2 = ( 2 + √ 3 + 2 - √ 3) 2 - 2 = (4) 2 - 2 = 16 - 2 = 14

Question 7

If m and n are natural numbers, the m√n

Accepted Answer

If m and n are natural numbers, then m√n is irrational unless n is mth power of an integer.

Question 8

If m = 7 - 4√3, then (√m + 1/√m) =?

Accepted Answer

Given that , m = 7 - 4√3 Then I/m = 1/7 - 4√3......................(1) Now multiply and divide with 7 + 4√3 in Eq. (1) We will get, 1/m = ( 1/7 - 4√3 ) x ( 7 + 4√3 / 7 + 4√3) ⇒1/m = ( 7 + 4√3 )/ ( 7 - 4√3 ) x ( 7 + 4√3 ⇒1/m = ( 7 + 4√3 ) / ( 72 - ( 4√3)2 ) ⇒1/m = ( 7 + 4√3 ) / ( 49 - 4 x 4 x 3 ) ⇒1/m = ( 7 + 4√3 ) /49 - 48 ⇒1/m = ( 7 + 4√3 ) / 1 ⇒1/m = 7 + 4√3 ∴ m + 1/m = 14 ⇒ ( m + 1/m ) + 2 = 14 + 2 = 16 ∴ (√m + 1/√m)2 = ( m + 1/m )+ 2 ⇒ (√m + 1/√m)2 = 16 ⇒ (√m + 1/√m)2 = 42 ⇒ (√m + 1/√m) = 4

Question 9

If √2n = 64 , what will be the value of n?

Accepted Answer

√2n = 64 ⇒ 2n = (64)2 ⇒ 2n = (2 × 2 × 2 × 2 × 2 × 2)2 ⇒ 2n = (26)2 ⇒ 2n = 26 × 2 = 212 ⇒ n = 12.

Question 10

(100)0 is equivalent to :

Accepted Answer

If any given number (suppose x) is a rational number other than zero, then x0 = 1.

Question 11

The value of 6a3 b3 c2 ÷ 2ab2 c is :

Accepted Answer

Question 12

(16)–3/4 + 2–3 + (8)–2/3 is equivalent to :

Accepted Answer

Question 13

[(2a2)3]3 [(3a3)2]2 ÷ [(6a6)2]2 is equivalent to :

Accepted Answer

Question 14

[(3 -2) -5] 1/5 + [ (4 -3) -6] 1/6 - 1 is equivalent to : [ ( 2-3 )-4 ]1/4

Accepted Answer

Question 15

∛ x6 ÷ ∛ x12 × x - 3 × ∛ x9 is equivalent to :

Accepted Answer

Question 16

If 3x - 3x-1 = 18, then xx is equal to

Accepted Answer

∵ 3x - 3x - 1 = 18 ⇒ 3x - 1(3 - 1) = 18 ⇒ 3x - 1(2) = 18 ⇒ 3x - 1 = 18/2 ⇒ 3x - 1 = 9 ⇒ 3x - 1 = 32 Apply the Algebra law, If aX = aY then X will be equal to Y. means X = Y; ⇒ x - 1 = 2 ⇒ x = 3 Then xx = (3)3 = 27

Question 17

If a2x + 2 = 1,where a is a positive real number other than 1, then x =?

Accepted Answer

Given that, a2x + 2 = 1 Apply the algebra Law, p0 = 1 ⇒ a2x + 2 = a0 Apply the algebra law pX = pY then X will be equal to Y. means X = Y; ⇒ 2x + 2 = 0 ⇒ x = -2/2 = -1

Question 18

Value of ? in expression 78.9 ÷ (343)1.7 x (49)4.8 = 7? is

Accepted Answer

78.9 ÷ (343)1.7 x (49)4.8 = 7? Apply the law of Fractional Exponents and Laws of Exponents (am)(an) = am+n am÷an=am?n am/an=am?n (am)n = amn 78.9 ÷ (343)1.7 x (49)4.8 = 7? ⇒ 78.9 ÷ (73)1.7 x (72)4.8 = 7? ⇒ 78.9 ÷ 75.1 x 79.6 = 7? ⇒ 78.9 - 5.1 + 9.6 = 7? ⇒ 718.5 - 5.1 = 7? ∴ ? = 13.4

Question 19

If √10 + ∛ x = 4, then what is the value of x?

Accepted Answer

Given, √10 + ∛ x = 4 On squaring both sides, we get 10 + ∛x = 16 ⇒ ∛x = 6 Now, cubic on both sides, we get x = (6)3 = 216

Question 20

If {(24)1/2}? = 256, find the value of '?'.

Accepted Answer

Given that, (24x 1/2)? = 256 Apply the law of Fractional Exponents and Laws of Exponents ⇒ (22)? = 28 ⇒ 22 x ? = 28 ⇒ 2 x ? = 8 ∴ ? = 8/2 = 4

Surds and Indices Questions