Sunday 25 December 2016

SURDS AND INDICES: Tricks Formulas Question answer pdf download

Here is in this article we are sharing Important quiz on surds and indices topic of Quantitative Aptitude subject. It is recommended for SSC, IBPS, SBI, Clerk and PO online exams. Top 14 example of surds indices aptitude questions are listed below. Generally this part question asked in most of all competitive exams, And also repeat many question. So interested candidates can study these question and revise the practice.

SURDS AND INDICES

I IMPORTANT FACTS AND FORMULAE I

1. LAWS OF INDICES:

(i) a^m x aⁿ = a^m^{+ n}

(ii) a^m / aⁿ = a^m-n

(iii) (a^m)ⁿ = a^mn

(iv) (ab)ⁿ = aⁿbⁿ

(v) ( a/ b )ⁿ = ( aⁿ / bⁿ )

(vi) a⁰ = 1

2. SURDS: Let a be a rational number and n be a positive integer such that a^1/n = ⁿsqrt(a)

is irrational. Then ⁿsqrt(a) is called a surd of order n.

3. LAWS OF SURDS:

(i) ⁿ√a = a^1/2

(ii) ⁿ√ab = ⁿ√a * ⁿ√b

(iii) ⁿ√a/b = ⁿ√a / ⁿ√b

(iv) (ⁿ √a)ⁿ = a

(v) ^m√(ⁿ√(a)) = ^mn√(a)

(vi) (ⁿ√a)^m = ⁿ√a^m

I SOLVED EXAMPLES

Question. 1. Simplify : (i) (27)^2/3 (ii) (1024)^-4/5 (iii)( 8 / 125 )^-4/3

Answer .
(i) (27)^2/3 = (3³)^2/3 = 3^{( 3 * ( 2/ 3))} = 3² = 9

(ii) (1024)^-4/5 = (4⁵)^-4/5= 4 ^{{ 5 * ( (-4) / 5 )}} = 4^-4 = 1 / 4⁴ = 1 / 256

(iii) ( 8 / 125 )^-4/3 = {(2/5)³}^-4/3 = (2/5)^{{ 3 * ( -4/3)}} = ( 2 / 5 )^-4 = ( 5 / 2 )⁴ = 5⁴/ 2⁴ = 625 / 16

Question. 2. Evaluate: (i) (.00032)^3/5 (ii)l (256)^0.16x (16)^0.18.

Answer .
(i) (0.00032)^3/5 = ( 32 / 100000 )^3/5. = (2⁵ / 10⁵)^3/5 = {( 2 / 10 )⁵}^3/5 = ( 1 / 5 )^{(5 * 3 / 5)} = (1/5)³ = 1 / 125

(ii) (256)^{0. 16} * (16)^{0. 18} = {(16)²}^{0. 16} * (16)^{0. 18} = (16)^{(2 * 0. 16)} * (16)^{0. 18}

=(16)^0.32 * (16)^0.18 = (16)^(0.32+0.18) = (16)^0.5 = (16)^1/2 = 4.

Question. 3. What is the quotient when (x^-1 - 1) is divided by (x - 1) ?

Answer .
x^-1 -1 = (1/x)-1 = _1 -x * 1 = -1

x - 1 x - 1 x (x - 1) x

Hence, the required quotient is -1/x

Question. 4. If 2^{x - 1} + 2^{x + 1} = 1280, then find the value of x.

Answer .
2^{x - 1} + 2^X+¹ = 1280 ó 2^x-1 (1 +2²) = 1280

ó 2^x-1 = 1280 / 5 = 256 = 2⁸ ó x -1 = 8 ó x = 9.

Hence, x = 9.

Question. 5. Find the value of [ 5 ( 8^1/3 + 27^1/3)³]^{1/ 4}

Answer .
[ 5 ( 8^1/3 + 27^1/3)³]^{1/ 4} = [ 5 { (2³)^1/3 + (3³)^1/3}³]^{1/ 4} = [ 5 { (2^{3 * 1/3})^1/3 + (3^{3 *1/3})^1/3}³]^{1/ 4}

= {5(2+3)³}^1/4 = (5 * 5³)^{1/ 4} =5^{(4 * 1/ 4)} = 5¹ = 5.

Question. 6. Find the Value of {(16)^3/2 + (16)^-3/2}

Answer .
[(16)^3/2 +(16)^-3/2 = (4²)^3/2 +(4²)^-3/2 = 4^{(2 * 3/2)} + 4^{{ 2* (-3/2)}}

= 4³ + 4^-3 = 4³ + (1/4³) = ( 64 + ( 1/64)) = 4097/64.

Question. 7. If (1/5)^3y = 0.008, then find the value of(0.25)^y.

Answer .
(1/5)^3y = 0.008 = 8/1000 = 1/125 = (1/5)³ ó 3y = 3 ó Y = 1.

\ (0.25)^y = (0.25)¹ = 0.25.

Question. 8. Find the value of (243)^n/5´ 3²ⁿ^{+ 1}9ⁿ´ 3ⁿ^-1 .

Answer .
(243)^n/5 x3^2n+l = 3 ^{(5 * n/5)} ´ 3^2n+l _ = 3ⁿ´3²ⁿ⁺¹

(3²)ⁿ ´ 3ⁿ^{- 1} 3²ⁿ ´ 3ⁿ^{- 1} 32n ´ 3^n-l

= 3ⁿ^{+ (2n + 1)} = 3⁽³ⁿ⁺¹⁾ = 3^{(3n+l)-(3n-l)} = 3² = 9.

3^2n+n-1 3^(3n-1)

Question. 9. Find the value Of (2^1/4-1)(2^3/4+2^1/2+2^1/4+1)

Answer .
Putting 2^1/4 = x, we get :

(2^1/4-1) (2^3/4+2^1/2+2^1/4+1)=(x-1)(x³+x²+x+1) , where x = 2^1/4

=(x-1)[x²(x+1)+(x+1)]

=(x-1)(x+1)(x²+1) = (x²-1)(x²+1)

=(x⁴-1) = [(2^1/4)⁴-1] = [2^(1/4*⁴⁾ –1] = (2-1) = 1.

Question. 10. Find the value of 6^2/3 ´ ³√6⁷

³√6⁶

Answer .
6^2/3 ´ ³√6⁷ = 6^2/3 ´ (6⁷)^1/3 = 6^2/3 ´ 6^{(7 * 1/3)} = 6^2/3 ´ 6^(7/3)

³√6⁶ (6⁶)^1/3 6^{(6 * 1/3)} 6²

=6^2/3 ´ 6^((7/3)-2) = 6^2/3 ´ 6^1/3 = 6¹= 6.

Question. 11. If x= y^a, y=z^b and z=x^c,then find the value of abc.

Answer .
z¹= x^c =(y^a)^c [since x= y^a]

=y^(ac) = (z^b)^ac [since y=z^b]

=z^b(ac)= z^abc

\ abc = 1.

= 24

Question. 12. Simplify [(x^a / x^b)^(a²+b²+ab)] * [(x^b / x^c )^ b²+c²+bc)] * [(x^c/x^a)^(c²+a²+ca)]

Answer .
Given Expression

= [{x^(o^{- b)}}^(a² + b² + ob)].['(x^(b^{- c)}}^ (b² + c² + bc)].['(x^(c^{- a)}}^(c² + a² + ca])

= [x^(a^{- b)(a2 + b2 + ab)} . x^(b^{- c) (b2 +c2+ bc)}.x^(c^{- a) (c2 + a2 + ca)}]

= [x^(a³-b³)].[x^(b³-e³)].[x^(c³-a³)] = x^(a³-b³+b³-c³+c³-a³) = x⁰ = 1.

Question. 13. Which is larger √2 or ³√3 ?

Answer .
Given surds are of order 2 and 3. Their L.C.M. is 6. Changing each to a surd of order 6, we get:

√2 = 2^1/2 = 2^{((1/2)*(3/2))} =2^3/6= 8^1/6 = ⁶√8

³√3= 3^1/3 = 3^{((1/3)*(2/2))} = 3^2/6 = (3²)^1/6 = (9)^1/6 =⁶√9.

Clearly, ⁶√9 > ⁶√8 and hence ³√3 > √2.

Question. 14. Find the largest from among 4√6, √2 and ³√4.

Answer .
Given surds are of order 4, 2 and 3 respectively. Their L.C,M, is 12, Changing each to a surd of order 12, we get:

⁴√6 = 6^1/4 = 6^{((1/4)*(3/3))} = 6^3/12 = (6³)^1/12 = (216)^1/12.

√2 = 2^1/2 = 2^{((1/2)*(6/6))} = 2^6/12 = (2⁶)^1/12 = (64)^1/12.

³√4 = 4^1/3 = 4^{((1/3)*(4/4))} = 4^4/12 = (4⁴)^1/12 = (256)^1/12.

Clearly, (256)^1/12 > (216)^1/12 > (64)^1/12

Largest one is (256)^1/12. i.e. ³√4 .

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General Knowledge One Liner Questions Answers

Important General Knowledge One Liner Questions and Answers for Bank Exam, SBI IBPS, SSC Clerk, PO and Govt. jobs Exam.

Q. 1. Alpha-keratin is a protein present in which thing?

Ans. Wool

Q. 2. Shovna Narayan is a reputed personality in which field?

Ans. Classical dance

Q. 3. Which post does the Constitution of India does not mention?

Ans. The Deputy Prime Minister

Q. 4. Which rocks is transformed into marble

Ans. Limestone

Q. 5. By which ruler was the practice of military governorship first introduced in India?

Ans. Greeks

Q. 7. The book ‘Diplomatic Encounter’ has been written by which author

Ans. Arundhati Roy

Q. 8. Which transition metal is in liquid state?

Ans. Mercury

Q. 9. What is the term as the drainage pattern developed on folded sedimentary rocks?

Ans. Trellis

Q. 10. Who was the first Indian woman President of the Indian National Congress?

Ans. Sarojini Naidu

We will update more GK Quiz and Notes on this blog.

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