Question 1

Evaluate ( 0.00032)2/5

Accepted Answer

Given that (0.00032)2/5 = (32/100000)2/5 Solve the equation by algebra Law = (25/105)2/5 = {(2/10)5}2/5 = (2/10)5x2/5 = (1/5)2 = 1/25

Question 2

If 2x - 1 + 2x + 1 = 2560, find the value of x.

Accepted Answer

2x - 1 + 2x + 1 = 2560 2x - 1 + 2x - 1 + 2 = 2560 Apply the Law of Algebra 2x - 1 + 2x - 1 x 2 2 = 2560 ⇒ 2x - 1 ( 1 + 22 ) = 2560 ⇒ 2x - 1 = 2560/5 = 512 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 ⇒ 2x - 1 = 29 ⇒ x - 1 = 9 ∴ x = 9 + 1 = 10

Question 3

Find the value of (10)200 ÷ (10)196 .

Accepted Answer

Given equation is (10)200 ÷ (10)196 Apply the law of Algebra am ÷ an = am ? n = (10)200 - 196 = 104 = 10000

Question 4

Value of √5 + 2√6 - 1/√5 - 2√6 is

Accepted Answer

√5 + 2√6 - 1/√5 - 2√6 = ((√5 + 2√6 x √5 - 2√6 ) - 1)/√5 - 2√6 Apply the formula of Algebra. ( a + b ) x (a - b ) = a2 - b2 = (√(5)2 - (2√6 )2 - 1)/√5 - 2√6 = (√25 - 4 x 6 - 1) /√5 - 2√6 = (√25 - 24 - 1) /√5 - 2√6 = (√ 1 - 1 )/√5 - 2√6 = (1 - 1 )/√5 - 2√6 = 0/√5 - 2√6 = 0

Question 5

If 2 x - 1 + 2 x + 1 = 320, then find the value of x?

Accepted Answer

∵ 2x - 1 + 2x + 1 = 320 Apply the law of Algebra ⇒ 2x - 1(1 + 2 2 ) = 320 ⇒ 2x - 1(1 + 4 ) = 320 ⇒ 2x - 1 x 5 = 320 ⇒ 2x - 1 = 64 = 2 x 2 x 2 x 2 x 2 x 2 ⇒ 2x - 1 = 26 if pX = pY then X will be equal to Y. means X = Y; ⇒ x - 1 = 6 ∴ x = 7

Question 6

if ax = b, by= c and xyz = 1, then what is the value of cz?

Accepted Answer

Given, xyz = 1, ax = b, by = c Now, b = ax Apply the law of algebra . Take the power y of both side equation. ⇒ by= axy Take the power z of both side equation. ⇒ byz = axyz ⇒ ( by )z = axyz Replace the by with c and xyz with 1 as per given equation, ⇒ cz = a

Question 7

If 16 x 8n + 2 = 2m, then m is equal to

Accepted Answer

Given that 16 x 8n + 2 = 2m Apply the law of Fractional Exponents and Laws of Exponents if a multiply three times a x a x a then a x a x a = a3 if a multiply two times a x a then a x a = a2 if a multiply n times a x a x a x....up to n times, then a x a x a x a ......up to n times = an ⇒ (2)4 x 23 x (n+2) = 2m ⇒ (2)4 x 23n+6 = 2m aman = am+n ⇒ (2)(4 + 3n + 6) = 2m ⇒ (2)(3n + 10) = 2m if pX = pY then X will be equal to Y. means X = Y; On comparing, we get 3n + 10 = m ⇒ m = 3n + 10;

Question 8

√56 + √56 + √56 + .... ) ÷ 22 =?

Accepted Answer

Here, Let us Assume , p = √56 + √56 + √56 + ........... ⇒ p = √56 + p Square on both side and solve the above equation. ⇒ p2= 56 + p ⇒ p2 - p= 56 ⇒ p(p - 1 ) = 56 ⇒ p(p - 1 ) = 8 x 7 ⇒ Either p = 8 or p - 1 = 7 ⇒ So p = 8; Now given equation in question is √56 + √56 + √56 + .... ) ÷ 22 = ? ⇒ p ÷ 22 = ? Put the value of p ⇒ 8 ÷ 22 = ? ⇒ 8 ÷ 4 = ? ⇒ ? = 2;

Question 9

If x = √3 + 1/ √3 - 1 and y = √3 - 1/ √3 + 1, then x2 + y2 is equal to

Accepted Answer

You can square the both the equation and add them to find the answer. x2 + y2 = ( (√3 + 1 ) / ( √3 - 1 ) )2 + ( (√3 - 1 ) / (√3 + 1) )2 x2 + y2 = ( √3 + 1 )2 / ( √3 - 1 ) 2 + (√3 - 1 )2 / (√3 + 1) 2 x2 + y2 = ( ( 3 + 1 + 2 √3 ) / ( 3 + 1 - 2 √3 ) ) + ( ( 3 + 1 - 2√3) / ( 3 + 1 + 2 √3 ) ) x2 + y2 = ( 4 + 2 √3 ) / ( 4 - 2 √3 ) + ( 4 - 2√3 ) / ( 4 + 2 √3 ) x2 + y2 = ( 2 + √3 ) / ( 2 - √3 ) + ( 2 - √3 ) / ( 2 + √3 ) x2 + y2 = (4 + 3 + 4 √3 + 4 + 3 - 4 √3) / ( 4 - 3 ) x2 + y2 = 14

Question 10

Find the value of m - n, if ( 9n x 32 x (3-n/2)-2 - (27)n ) / ( 33m x 23 ) = 1/27?

Accepted Answer

( 9n x 32 x (3-n/2)-2 - (27)n ) / ( 33m x 23 ) = 1/27 ⇒ ( 32n x 32 x 3n - 33n) / ( 33m x 23 ) = 1/27 ⇒ ( 32n + n x 32 - 33n) / ( 33m x 23 )= 1/27 ⇒ ( 33n x 32 - 33n) / ( 33m x 23 ) = 1/27 ⇒ 33n( 32 - 1) / ( 33m x 8 ) = 1/27 ⇒ 33n( 9 - 1) / (33m x 8 ) = 1/27 ⇒ ( 33n x 8 ) / (33m x 8 ) = 1/27 ⇒ 33n / 33m = 1/27 ⇒ 33n - 3m = 1/33 ⇒ 33( n - m ) = 3-3 3(n - m) = -3 or n - m = -1 or m - n = 1

Question 11

Find the quotient when (a-1 - 1) is divided by (a - 1).

Accepted Answer

Given equation is a-1 - 1 Apply the law of Fractional Exponents and Laws of Exponents x-y = 1/xy ⇒ a-1 - 1 = 1/a - 1 ⇒ a-1 - 1 = (1 - a)/a Now divide by a - 1 in above equation, ⇒ ( a-1 - 1 ) ÷ (a - 1) = (1 - a)/a ÷ (a - 1) ⇒ ( a-1 - 1 ) ÷ (a - 1) = (1 - a)/a x 1/ (a - 1) ⇒ ( a-1 - 1 ) ÷ (a - 1) = -1x (a - 1)/a x 1/ (a - 1) ⇒ ( a-1 - 1 ) ÷ (a - 1) = -1/a ∴ Required quotient = -1/a.

Question 12

What is the difference between the third power of 2 and the second power of 3?

Accepted Answer

Third power of 2 = 23 = 2 × 2 × 2 = 8 And second power of 3 = 32 = 3 × 3 = 9 ∴ Difference = 9 – 8 = 1

Question 13

What is the value of a5 × a7?

Accepted Answer

a5 × a7 = a5 + 7 = a12

Question 14

Third power of 4 is equivalent to :

Accepted Answer

Third power of 4 = 43 = 4 × 4 × 4 = 64

Question 15

( x2/3)- 3/4 is equivalent to :

Accepted Answer

Question 16

Solve the below equation. 173.5 x 177.3 ÷ 174.2 = 17?

Accepted Answer

173.5 x 177.3 ÷ 174.2 = 17? Apply the Laws of Exponents :: (am) x (an) = am+n Fractional Exponents :: am÷ a n=am?n ⇒ 173.5 + 7.3 - 4.2 = 17? ⇒ 176.6 = 17? ∴ ? = 6.6

Question 17

If 289 = 17x/5 , then x =?

Accepted Answer

Given that , 289 = 17x/5 Use the formula if ax = a y then x = y will be true. ⇒ 171/2 = 17x/5 ⇒ x /5 = 1/2 ⇒ x = 10 .

Question 18

L√M is surd of order ........, where M is a rational number; L is a positive integer and L√M is irrational.

Accepted Answer

L√3 = M1/L = Surd of Lth order.

Question 19

If 2p + 3q = 17 and 2p + 2 - 3q + 1 = 5, then, find the value of p and q?

Accepted Answer

Apply the law of Fractional Exponents and Laws of Exponents (am) x (an) = am + n a-m = 1/am p0 = 1 and if pX = pY then X will be equal to Y. means X = Y; (am)n = am x n (am) x (an) = am + n am ÷ an = am ? n Or am/an = am ? n Given equation is 2p + 3q = 17 ----------------(i) 2p + 2 - 3q + 1 = 5 ⇒ 2p x 22 - 3q x 3 = 5 ⇒ 2p x 4 - 3q x 3 = 5 ⇒ 4 x 2p - 3 x 3q = 5---------------(ii) On multiplying Eq. (i) by 3 and adding it, with Eq. (ii), we get 3 x 2p + 3 x 3q = 51 4 x 2p - 3 x 3q = 5 _____________________________ 7 x 2p = 56 ⇒ 7 x 2p = 7 x 8 ⇒ 2p = 8 ⇒ 2p = 23 ⇒ p = 3 On substituting the value of p in Eq. (i) . we get 23 + 3q = 17 ⇒ 8 + 3q = 17 ⇒ 3q = 17 - 8 ⇒ 3q = 9 ⇒ 3q = 32 ⇒ q = 2

Question 20

( 42 x 229 ) ÷ ( 9261)1/3 =?

Accepted Answer

Given that, (42 x 229) ÷ (9261)1/3 = ? As we know that (9621)1/3 = (21 x 21 x 21)1/3 = 213x1/3 ∴ 42 x 229 ÷ (213 x )1/3 = ? ∴ 42 x 229 ÷ 21 = ? ∴ 42 x 229 x 1/ 21 = ? ∴ 2 x 229 = ? ∴ ? = 2 x 229 ∴ ? = 458

Surds and Indices Questions