194 + Interview Questions in EXPONENTS IN GENERAL AWARENESS Page 2 InterviewSolution

51.	If 64a = \(\frac{1}{256^b}\), then 3a+4b equals(a) 2 (b) 4 (c) 8 (d) 0
Answer» (d) 0 64^a = \(\frac{1}{256^b}\) ⇒ (2⁶)^a= \(\frac{1}{(2^8)^b}\) ⇒ 2^6a × 2^8b = 1 ⇒ 2^6a ^{+ 8b} = 2⁰ ⇒ 6a + 8b = 0 ⇒ 3a + 4b = 0

51.

If 64a = \(\frac{1}{256^b}\), then 3a+4b equals(a) 2 (b) 4 (c) 8 (d) 0

Answer»

(d) 0

64^a = \(\frac{1}{256^b}\) ⇒ (2⁶)^a= \(\frac{1}{(2^8)^b}\)

⇒ 2^6a × 2^8b = 1 ⇒ 2^6a ^{+ 8b} = 2⁰

⇒ 6a + 8b = 0 ⇒ 3a + 4b = 0

Discussion

52.	(1/2)-2 + (1/3)-2 + (1/4)-2 = ?(a) (61/144) (b) 29 (c) (144/61) (d) none of these
Answer» (b) 29 We know that, = (1/2)^-2= (2/1)²… [∵ (a/b)^-n = (b/a)ⁿ] = (1/3)^-2 = (3/1)² … [∵ (a/b)^-n = (b/a)ⁿ] = (1/4)^-2 = (4/1)² … [∵ (a/b)^-n = (b/a)ⁿ] Now add, = (2)²+ (3)²+ (4)² = 4 + 9 + 16 = 29

52.

(1/2)-2 + (1/3)-2 + (1/4)-2 = ?(a) (61/144) (b) 29 (c) (144/61) (d) none of these

Answer»

(b) 29

We know that,

= (1/2)^-2= (2/1)²… [∵ (a/b)^-n = (b/a)ⁿ]

= (1/3)^-2 = (3/1)² … [∵ (a/b)^-n = (b/a)ⁿ]

= (1/4)^-2 = (4/1)² … [∵ (a/b)^-n = (b/a)ⁿ]

Now add,

= (2)²+ (3)²+ (4)²

= 4 + 9 + 16

= 29

Discussion

53.	43.5 : 25 is the same as(a) 4 : 1 (b) 2 : 1 (c) 7 : 5 (d) 7 : 10
Answer» (a) 4 : 1 4^3.5 : 2⁵ = (2²)^3.5 : 2⁵ = 2⁷ : 2⁵ = \(\frac{2^7}{2^5}\) : 1 = 2^7–5 : 1 = 2² : 1 = 4 : 1.

53.

43.5 : 25 is the same as(a) 4 : 1 (b) 2 : 1 (c) 7 : 5 (d) 7 : 10

Answer»

(a) 4 : 1

4^3.5 : 2⁵ = (2²)^3.5 : 2⁵ = 2⁷ : 2⁵

= \(\frac{2^7}{2^5}\) : 1 = 2^7–5 : 1 = 2² : 1 = 4 : 1.

Discussion

54.	{6-1+(3/2)-1}-1(a) (2/3) (b) (5/6) (c) (6/5) (d) None of these
Answer» (c) (6/5) We know that, = (6)^-1= (1/6) … [∵ (a/b)^-n = (b/a)ⁿ] = (3/2)^-1 = (2/3) … [∵ (a/b)^-n = (b/a)ⁿ] Now add, = {(1/6) + (2/3)}^-1 = {(1+4)/ 6}^-1 … [LCM of 6 and 3 is 6] = {5/6}^-1 = {6/5}

54.

{6-1+(3/2)-1}-1(a) (2/3) (b) (5/6) (c) (6/5) (d) None of these

Answer»

(c) (6/5)

We know that,

= (6)^-1= (1/6) … [∵ (a/b)^-n = (b/a)ⁿ]

= (3/2)^-1 = (2/3) … [∵ (a/b)^-n = (b/a)ⁿ]

Now add,

= {(1/6) + (2/3)}^-1

= {(1+4)/ 6}^-1 … [LCM of 6 and 3 is 6]

= {5/6}^-1

= {6/5}

Discussion

55.	{(3/4)-1 – (1/4)-1}-1 = ?(a) (3/8) (b) (-3/8) (c) (8/3) (d) (-8/3)
Answer» (b) (-3/8) We know that, = (3/4)^-1= (4/3)¹… [∵ (a/b)^-n = (b/a)ⁿ] = (1/4)^-1 = (4/1)¹ … [∵ (a/b)^-n = (b/a)ⁿ] Now subtract, = {(4/3) – (4/1)}^-1 = {(4-12)/3}^-1 … [LCM of 3 and 1 is 3] = {-8/3}^-1 = {-3/8}

55.

{(3/4)-1 – (1/4)-1}-1 = ?(a) (3/8) (b) (-3/8) (c) (8/3) (d) (-8/3)

Answer»

(b) (-3/8)

We know that,

= (3/4)^-1= (4/3)¹… [∵ (a/b)^-n = (b/a)ⁿ]

= (1/4)^-1 = (4/1)¹ … [∵ (a/b)^-n = (b/a)ⁿ]

Now subtract,

= {(4/3) – (4/1)}^-1

= {(4-12)/3}^-1 … [LCM of 3 and 1 is 3]

= {-8/3}^-1

= {-3/8}

Discussion

56.	(-1/2)-6 = ?(a) -64 (b) 64 (c) (1/64) (d) (-1/64)
Answer» (b) 64 We know that, = (-1/2)^-6= (-2/1)⁶… [∵ (a/b)^-n = (b/a)ⁿ] = (-2)⁶ = 64

56.

(-1/2)-6 = ?(a) -64 (b) 64 (c) (1/64) (d) (-1/64)

Answer»

(b) 64

We know that,

= (-1/2)^-6= (-2/1)⁶… [∵ (a/b)^-n = (b/a)ⁿ]

= (-2)⁶

= 64

Discussion

57.	(2/3)0 = ?A. 3/2B. 2/3C. 1D. 0
Answer» By using the law of exponents \((\frac{a}{b})^0=1\) \(\therefore(\frac{2}{3})^0=1\)

57.

(2/3)0 = ?A. 3/2B. 2/3C. 1D. 0

Answer»

By using the law of exponents \((\frac{a}{b})^0=1\)

\(\therefore(\frac{2}{3})^0=1\)

Discussion

58.	(-5/3)-1 = 1A. 5/3B. 3/5C. -3/5D. None of these
Answer» \((\frac{-5}{3})^{-1}=\frac{1}{\frac{5}{3}}=-\frac{3}{5}\)

Discussion

59.	(-1/2)3 = ?A. -1/6B. 1/6C. 1/8D. -1/8
Answer» \((-\frac{1}{2})^3=-\frac{1}{2}\times-\frac{1}{2}=-\frac{1}{8}\)

Discussion

60.	(-3/4)2 = ?A. -9/16B. 9/16C. 16/9D. -16/9
Answer» \((-\frac{3}{4})^2=-\frac{3}{4}\times-\frac{3}{4}=\frac{9}{16}\)

Discussion

61.	[{(-1/2)2}-2]-1=?(a) (1/16) (b) 16 (c) (-1/16) (d) -16
Answer» (a) (1/16) [{(-1/2)²}^-2]^-1= [{(-1²/2²)}^-2]^-1 = [{1/4}^-2]^-1 = [{4}²]^-1 = [16^]-1 = [1/16]

61.

[{(-1/2)2}-2]-1=?(a) (1/16) (b) 16 (c) (-1/16) (d) -16

Answer»

(a) (1/16)

[{(-1/2)²}^-2]^-1= [{(-1²/2²)}^-2]^-1

= [{1/4}^-2]^-1

= [{4}²]^-1

= [16^]-1

= [1/16]

Discussion

62.	Find the reciprocal of (-5/6)11.
Answer» (-5/6)¹¹ We know that the reciprocal of (a/b)^m is (b/a) ^m Then, Reciprocal of (-5/6)¹¹ is (-6/5)¹¹

62.

Find the reciprocal of (-5/6)11.

Answer»

(-5/6)¹¹

We know that the reciprocal of (a/b)^m is (b/a) ^m

Then,

Reciprocal of (-5/6)¹¹ is (-6/5)¹¹

Discussion

63.	Express the following information in the standard form. (i) The size of red blood cells is 0.000007mm(ii) The speed of light is 300000000 m/sec(iii) The distance between the moon and the earth is 384467000 m(app)(iv) The charge of an electron is 0.0000000000000000016 coulombs(v) Thickness of a piece of paper is 0.0016 cm(vi) The diameter of a wire on a computer chip is 0.000005 cm
Answer» (i) The size of red blood cells is 0.000007mm = 7 / 1000000 = 7 × 10^-6 (ii) The speed of light is 300000000 m/sec = 3 × 10,00,00,000 = 3 × 10⁸ m/sec (iii) The distance between the moon and the earth is 384467000 m(app) = 384467 × 1000 m = 384467 × 10³ (iv) The charge of an electron is 0.0000000000000000016 coulombs = 0.0000000000000000016 = 16 / 10000000000000000000 = 16 / 10¹⁹ = 16 × 10^-19 coulombs (v) Thickness of a piece of paper is 0.0016 cm = 0.0016 cm = 16 / 10000 = 16 / 10⁴ = 16 × 10^-4 cm (vi) The diameter of a wire on a computer chip is 0.000005 cm = 0.000005 cm = 5 / 1000000 cm = 5 / 10⁶ cm = 5 × 10^-6 cm

63.

Express the following information in the standard form. (i) The size of red blood cells is 0.000007mm(ii) The speed of light is 300000000 m/sec(iii) The distance between the moon and the earth is 384467000 m(app)(iv) The charge of an electron is 0.0000000000000000016 coulombs(v) Thickness of a piece of paper is 0.0016 cm(vi) The diameter of a wire on a computer chip is 0.000005 cm

Answer»

(i) The size of red blood cells is 0.000007mm

= 7 / 1000000 = 7 × 10^-6

(ii) The speed of light is 300000000 m/sec

= 3 × 10,00,00,000 = 3 × 10⁸ m/sec

(iii) The distance between the moon and the earth is 384467000 m(app)

= 384467 × 1000 m

= 384467 × 10³

(iv) The charge of an electron is 0.0000000000000000016 coulombs

= 0.0000000000000000016

= 16 / 10000000000000000000

= 16 / 10¹⁹

= 16 × 10^-19 coulombs

(v) Thickness of a piece of paper is 0.0016 cm

= 0.0016 cm = 16 / 10000

= 16 / 10⁴

= 16 × 10^-4 cm

(vi) The diameter of a wire on a computer chip is 0.000005 cm

= 0.000005 cm = 5 / 1000000 cm

= 5 / 10⁶ cm = 5 × 10^-6 cm

Discussion

64.	What is 10-10 equal to?
Answer» 10^-10 = 1 / 10¹⁰ [∵ a^-n = 1 / aⁿ]

Discussion

65.	The stand form of the distance between the earth and the moon is aprox. 12,000,000,000 mts A) 12 × 1011 years B) 1.2 × 1010 years C) 0.12 × 109 years D) 120 × 1011 years
Answer» Correct option is B) 1.2 × 10¹⁰ years

Discussion

66.	Prime factorisation of 72 is ………….?A) 23 × 32B) 23 × 33C) 33 × 22 D) 2 × 32 × 7
Answer» Correct option is A) 2³ × 3²

Discussion

67.	(1/2)-2 + (1/3)-2 + (1/4)-2 = ?A. 61/144B. 144/61C. 29D. 1/29
Answer» \((\frac{1}{2})^{-2}+(\frac{1}{3})^{-2}+(\frac{1}{4})^{-2}\) = \((\frac{2}{1})^2+(\frac{3}{1})^2+(\frac{4}{1})^2\) = 2²+3²+4² = 4+9+16 = 29

67.

(1/2)-2 + (1/3)-2 + (1/4)-2 = ?A. 61/144B. 144/61C. 29D. 1/29

Answer»

\((\frac{1}{2})^{-2}+(\frac{1}{3})^{-2}+(\frac{1}{4})^{-2}\) = \((\frac{2}{1})^2+(\frac{3}{1})^2+(\frac{4}{1})^2\)

= 2²+3²+4²

= 4+9+16

= 29

Discussion

68.	Express each of the following numbers as a product of powers of their prime factors:(i) 450(ii) 2800(iii) 24000
Answer» (i) Given 450 Prime factorization of 450 = 2 x 3 x 3 x 5 x 5 = 2 x 3² x 5² (ii) Given 2800 Prime factorization of 2800 = 2 x 2 x 2 x 2 x 5 x 5 x 7 = 2⁴ x 5² x 7 (iii) Given 24000 Prime factorization of 24000 = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 5 x 5 x 5 = 2⁶ x 3 x 5³

68.

Express each of the following numbers as a product of powers of their prime factors:(i) 450(ii) 2800(iii) 24000

Answer»

(i) Given 450

Prime factorization of 450 = 2 x 3 x 3 x 5 x 5

= 2 x 3² x 5²

(ii) Given 2800

Prime factorization of 2800 = 2 x 2 x 2 x 2 x 5 x 5 x 7

= 2⁴ x 5² x 7

(iii) Given 24000

Prime factorization of 24000 = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 5 x 5 x 5

= 2⁶ x 3 x 5³

Discussion

69.	r × r × 7 times = ………………? A) 7rB) 7r C) r7D) None
Answer» Correct option is C) r⁷ r x r x....x r (7 times) = r⁷

69.

r × r × 7 times = ………………? A) 7rB) 7r C) r7D) None

Answer»

Correct option is C) r⁷

r x r x....x r (7 times) = r⁷

Discussion

70.	Express (5/7)-1× (7/4)-1 as a rational number.
Answer» (5/7)^-1× (7/4)^-1 We know that, = (5/7)^-1= (7/5)¹… [∵ (a/b)^-n = (b/a)ⁿ] = (7/4)^-1 = (4/7)¹ … [∵ (a/b)^-n = (b/a)ⁿ] = (7/5) × (4/7) = (7×4)/ (5×7) On simplifying, = (1×4)/ (5×1) = 4/5

70.

Express (5/7)-1× (7/4)-1 as a rational number.

Answer»

(5/7)^-1× (7/4)^-1

We know that,

= (5/7)^-1= (7/5)¹… [∵ (a/b)^-n = (b/a)ⁿ]

= (7/4)^-1 = (4/7)¹ … [∵ (a/b)^-n = (b/a)ⁿ]

= (7/5) × (4/7)

= (7×4)/ (5×7)

On simplifying,

= (1×4)/ (5×1)

= 4/5

Discussion

71.	Express (-6)-1 as a rational number.
Answer» (-6)^-1 We have: (-6)^-1 = (-6/1)^-1 = (1/-6)¹ … [∵ (a/b)^-n = (b/a)ⁿ] = (-1/6)

71.

Express (-6)-1 as a rational number.

Answer»

(-6)^-1

We have:

(-6)^-1 = (-6/1)^-1

= (1/-6)¹ … [∵ (a/b)^-n = (b/a)ⁿ]

= (-1/6)

Discussion

72.	If x = yz , y = zx and z = xy , then(a) \(\frac{xy}{z} = 1\)(b) xyz = 1 (c) x + y + z = 1 (d) xz = y
Answer» (b) xyz = 1 z = x^y = (y^z)^y (∵ x = y^z ) = y^zy = (z^x)^zy = z^xyz (∵ y = z^x) ∴ z¹ = z^xyz ⇒ xyz = 1

72.

If x = yz , y = zx and z = xy , then(a) \(\frac{xy}{z} = 1\)(b) xyz = 1 (c) x + y + z = 1 (d) xz = y

Answer»

(b) xyz = 1

z = x^y = (y^z)^y (∵ x = y^z )

= y^zy = (z^x)^zy = z^xyz (∵ y = z^x)

∴ z¹ = z^xyz ⇒ xyz = 1

Discussion

73.	Find the value of x if \(\big[3^{2x-2}+10\big]\) ÷ 13 = 7.(a) 1 (b) 3 (c) 4 (d) 2
Answer» (b) 3 \(\big[3^{2x-2}+10\big]\) ÷ 13 = 7. ⇒ 3^{2x – 2} + 10 = 7 × 13 = 91 ⇒ 3^{2x – 2} = 91 – 10 = 81 = 3⁴ ⇒ 2x –2 = 4 ⇒ 2x = 6 ⇒ x = 3.

73.

Find the value of x if \(\big[3^{2x-2}+10\big]\) ÷ 13 = 7.(a) 1 (b) 3 (c) 4 (d) 2

Answer»

(b) 3

\(\big[3^{2x-2}+10\big]\) ÷ 13 = 7.

⇒ 3^{2x – 2} + 10 = 7 × 13 = 91

⇒ 3^{2x – 2} = 91 – 10 = 81 = 3⁴

⇒ 2x –2 = 4 ⇒ 2x = 6 ⇒ x = 3.

Discussion

74.	Express (-2)-5 as a rational number.
Answer» (-2)^-5 We know that, = (-2)^-5 = (-1/2)³ … [∵ (a/b)^-n = (b/a)ⁿ] = (-1³/2³) = (-1/8)

74.

Express (-2)-5 as a rational number.

Answer»

(-2)^-5

We know that,

= (-2)^-5 = (-1/2)³ … [∵ (a/b)^-n = (b/a)ⁿ]

= (-1³/2³)

= (-1/8)

Discussion

75.	Express each of the following numbers as a product of powers of their prime factors:(i) 36(ii) 675(iii) 392
Answer» (i) Given 36 Prime factorization of 36 = 2 x 2 x 3 x 3 = 2² x 3² (ii) Given 675 Prime factorization of 675 = 3 x 3 x 3 x 5 x 5 = 3³ x 5² (iii) Given 392 Prime factorization of 392 = 2 x 2 x 2 x 7 x 7 = 2³ x 7²

75.

Express each of the following numbers as a product of powers of their prime factors:(i) 36(ii) 675(iii) 392

Answer»

(i) Given 36

Prime factorization of 36 = 2 x 2 x 3 x 3

= 2² x 3²

(ii) Given 675

Prime factorization of 675 = 3 x 3 x 3 x 5 x 5

= 3³ x 5²

(iii) Given 392

Prime factorization of 392 = 2 x 2 x 2 x 7 x 7

= 2³ x 7²

Discussion

76.	By what number should three be multiplied so that the product is 729’?
Answer» Given number = 3^-4 Given product = 729 [∵ 3⁶ = 729] Let the number to be multiplied be x then ⇒ (3^-4) . (x) = 3⁶ ⇒ x/3⁴ =3⁶ [∵ a^m × nⁿ = a^m+n] ⇒ x = 3⁶ × 3⁴ = 3¹⁰

76.

By what number should three be multiplied so that the product is 729’?

Answer»

Given number = 3^-4

Given product = 729 [∵ 3⁶ = 729]

Let the number to be multiplied be x then

⇒ (3^-4) . (x) = 3⁶ ⇒ x/3⁴ =3⁶

[∵ a^m × nⁿ = a^m+n]

⇒ x = 3⁶ × 3⁴ = 3¹⁰

Discussion

77.	Express each of the following numbers in exponential form:(i) 512(ii) 625(iii) 729
Answer» (i) Given 512 Prime factorization of 512 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 2⁹ (ii) Given 625 Prime factorization of 625 = 5 x 5 x 5 x 5 = 5⁴ (iii) Given 729 Prime factorization of 729 = 3 x 3 x 3 x 3 x 3 x 3 = 3⁶

77.

Express each of the following numbers in exponential form:(i) 512(ii) 625(iii) 729

Answer»

(i) Given 512

Prime factorization of 512

= 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2

= 2⁹

(ii) Given 625

Prime factorization of 625

= 5 x 5 x 5 x 5

= 5⁴

(iii) Given 729

Prime factorization of 729

= 3 x 3 x 3 x 3 x 3 x 3

= 3⁶

Discussion

78.	If a = 3, b = 2 find the value of(i) ab + ba(ii) aa + bb(iii) (a + b)ab(iv)(a – b)a
Answer» (i) a^b + b^a = 3² + 2³ = 3 × 3 + 2 × 2 × 2 = 9 + 8 =17 (ii) a^a + b^b = 3³ + 2² = 3 × 3 × 3 + 2 × 2 = 27 + 4 = 31 (iii) (a + b)^b = (3 + 2)² = 5² = 5 × 5 = 25 (iv)(a – b)^a = (3 – 2)²= 1² = 1 × 1 = 1 .

78.

If a = 3, b = 2 find the value of(i) ab + ba(ii) aa + bb(iii) (a + b)ab(iv)(a – b)a

Answer»

(i) a^b + b^a = 3² + 2³ = 3 × 3 + 2 × 2 × 2 = 9 + 8 =17

(ii) a^a + b^b = 3³ + 2² = 3 × 3 × 3 + 2 × 2 = 27 + 4 = 31

(iii) (a + b)^b = (3 + 2)² = 5² = 5 × 5 = 25

(iv)(a – b)^a = (3 – 2)²= 1² = 1 × 1 = 1 .

Discussion

79.	Express each of the following in exponential form:(i) x × x × x × x × a × a × b × b × b(ii) (-2) × (-2) × (-2) × (-2) × a × a × a(iii) (-2/3) × (-2/3) × x × x × x
Answer» (i) Given x × x × x × x × a × a × b × b × b Exponential form of x × x × x × x × a × a × b × b × b = x⁴a²b³ (ii) Given (-2) × (-2) × (-2) × (-2) × a × a × a Exponential form of (-2) × (-2) × (-2) × (-2) × a × a × a = (-2)⁴a³ (iii) Given (-2/3) × (-2/3) × x × x × x Exponential form of (-2/3) × (-2/3) × x × x × x = (-2/3)²x³

79.

Express each of the following in exponential form:(i) x × x × x × x × a × a × b × b × b(ii) (-2) × (-2) × (-2) × (-2) × a × a × a(iii) (-2/3) × (-2/3) × x × x × x

Answer»

(i) Given x × x × x × x × a × a × b × b × b

Exponential form of x × x × x × x × a × a × b × b × b = x⁴a²b³

(ii) Given (-2) × (-2) × (-2) × (-2) × a × a × a

Exponential form of (-2) × (-2) × (-2) × (-2) × a × a × a = (-2)⁴a³

(iii) Given (-2/3) × (-2/3) × x × x × x

Exponential form of (-2/3) × (-2/3) × x × x × x = (-2/3)²x³

Discussion

80.	Identify the greater number in each of the following pairs.(i) 23 or 32(ii) 53 or 35(iii) 28 or 82
Answer» (i) 2³ or 3² = 2³ = 2 × 2 × 2 = 8 and 3² = 3 × 3 = 9 ∴ 2³ < 3² or 3² > 2³ (ii) 5³ or 3⁵ = 5³ = 5 × 5 × 5 = 125 and 3⁵ = 3 × 3 × 3 × 3 × 3 = 243 ∴ 3⁵ > 5³ iii) 2⁸ or 8² = 2⁸ = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256 8² = 8 × 8 = 64 ∴ 2⁸ > 8²

80.

Identify the greater number in each of the following pairs.(i) 23 or 32(ii) 53 or 35(iii) 28 or 82

Answer»

(i) 2³ or 3² = 2³ = 2 × 2 × 2

= 8 and 3² = 3 × 3 = 9

∴ 2³ < 3² or 3² > 2³

(ii) 5³ or 3⁵ = 5³ = 5 × 5 × 5 = 125 and 3⁵

= 3 × 3 × 3 × 3 × 3 = 243

∴ 3⁵ > 5³

iii) 2⁸ or 8² = 2⁸

= 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256

8² = 8 × 8 = 64

∴ 2⁸ > 8²

Discussion

81.	Express each of the following in exponential form:(i) (-5) × (-5) × (-5)(ii) (-5/7) × (-5/7) × (-5/7) × (-5/7)(iii) (4/3) × (4/3) × (4/3) × (4/3) × (4/3)
Answer» (i) Given (-5) × (-5) × (-5) Exponential form of (-5) × (-5) × (-5) = (-5)³ (ii) Given (-5/7) × (-5/7) × (-5/7) × (-5/7) Exponential form of (-5/7) × (-5/7) × (-5/7) × (-5/7) = (-5/7)⁴ (iii) Given (4/3) × (4/3) × (4/3) × (4/3) × (4/3) Exponential form of (4/3) × (4/3) × (4/3) × (4/3) × (4/3) = (4/3)⁵

81.

Express each of the following in exponential form:(i) (-5) × (-5) × (-5)(ii) (-5/7) × (-5/7) × (-5/7) × (-5/7)(iii) (4/3) × (4/3) × (4/3) × (4/3) × (4/3)

Answer»

(i) Given (-5) × (-5) × (-5)

Exponential form of (-5) × (-5) × (-5) = (-5)³

(ii) Given (-5/7) × (-5/7) × (-5/7) × (-5/7)

Exponential form of (-5/7) × (-5/7) × (-5/7) × (-5/7) = (-5/7)⁴

(iii) Given (4/3) × (4/3) × (4/3) × (4/3) × (4/3)

Exponential form of (4/3) × (4/3) × (4/3) × (4/3) × (4/3) = (4/3)⁵

Discussion

82.	Identify the greater number in each of the following:(i) 25 or 52(ii) 34 or 43(iii) 35 or 53
Answer» (i) Given 2⁵ or 5² 2⁵ = 2 × 2 × 2 × 2 × 2 = 32 5² = 5 × 5 = 25 Therefore, 2⁵ > 5² (ii) Given 3⁴ or 4³ 3⁴ = 3 × 3 × 3 × 3 = 81 4³= 4 × 4 × 4 = 64 Therefore, 3⁴ > 4³ (iii) Given 3⁵or 5³ 3⁵ = 3 × 3 × 3 × 3 × 3 = 243 5³ = 5 × 5 × 5 = 125 Therefore, 3⁵ > 5³

82.

Identify the greater number in each of the following:(i) 25 or 52(ii) 34 or 43(iii) 35 or 53

Answer»

(i) Given 2⁵ or 5²

2⁵ = 2 × 2 × 2 × 2 × 2

= 32

5² = 5 × 5

= 25

Therefore, 2⁵ > 5²

(ii) Given 3⁴ or 4³

3⁴ = 3 × 3 × 3 × 3

= 81

4³= 4 × 4 × 4

= 64

Therefore, 3⁴ > 4³

(iii) Given 3⁵or 5³

3⁵ = 3 × 3 × 3 × 3 × 3

= 243

5³ = 5 × 5 × 5

= 125

Therefore, 3⁵ > 5³

Discussion

83.	Express (4)-1 as a rational number.
Answer» (4)^-1 We have: (4)^-1 = (4/1)^-1 = (1/4)¹… [∵ (a/b)^-n = (b/a)ⁿ] = (1/4)

83.

Express (4)-1 as a rational number.

Answer»

(4)^-1

We have:

(4)^-1 = (4/1)^-1

= (1/4)¹… [∵ (a/b)^-n = (b/a)ⁿ]

= (1/4)

Discussion

84.	Express (-1)9 a rational number.
Answer» (-1)⁹ We have, (-1)⁹ = (-1⁹) = (-1 × -1 × -1× -1 × -1 × -1 × -1× -1 × -1) = (-1)

84.

Express (-1)9 a rational number.

Answer»

(-1)⁹

We have,

(-1)⁹ = (-1⁹)

= (-1 × -1 × -1× -1 × -1 × -1 × -1× -1 × -1)

= (-1)

Discussion

85.	Express (-1/2)5 a rational number.
Answer» (-1/2)⁵ We have, (-1/2)⁵ = (-1⁵/2⁵) = (-1 × -1 × -1× -1 × -1) / (2 × 2 × 2 × 2 × 2) = (-1/32)

85.

Express (-1/2)5 a rational number.

Answer»

(-1/2)⁵

We have,

(-1/2)⁵ = (-1⁵/2⁵)

= (-1 × -1 × -1× -1 × -1) / (2 × 2 × 2 × 2 × 2)

= (-1/32)

Discussion

86.	Express (2/3)5 a rational number.
Answer» (2/3)⁵ We have, (2/3)⁵ = (2⁵/3⁵) = (2 × 2 × 2 × 2 × 2) / (3 × 3 × 3 × 3 × 3) = (32/243)

86.

Express (2/3)5 a rational number.

Answer»

(2/3)⁵

We have,

(2/3)⁵ = (2⁵/3⁵)

= (2 × 2 × 2 × 2 × 2) / (3 × 3 × 3 × 3 × 3)

= (32/243)

Discussion

87.	Express (-4/7)3 a rational number.
Answer» (-4/7)³ We have, (-4/7)³ = (-4³/7³) = (-4 × -4 × -4) / (7 × 7 × 7) = (-64/343)

87.

Express (-4/7)3 a rational number.

Answer»

(-4/7)³

We have,

(-4/7)³ = (-4³/7³)

= (-4 × -4 × -4) / (7 × 7 × 7)

= (-64/343)

Discussion

88.	Express (-8/5)3 a rational number.
Answer» (-8/5)³ We have, (-8/5)³ = (-8³/5³) = (-8 × -8 × -8) / (5 × 5 × 5) = (-512/125)

88.

Express (-8/5)3 a rational number.

Answer»

(-8/5)³

We have,

(-8/5)³ = (-8³/5³)

= (-8 × -8 × -8) / (5 × 5 × 5)

= (-512/125)

Discussion

89.	Express (1/6)3 a rational number.
Answer» (1/6)³ We have, (1/6)³ = (1³/6³) = (1 × 1 × 1) / (6 × 6 × 6) = (1/216)

89.

Express (1/6)3 a rational number.

Answer»

(1/6)³

We have,

(1/6)³ = (1³/6³)

= (1 × 1 × 1) / (6 × 6 × 6)

= (1/216)

Discussion

90.	Express the following numbers in the exponential form.i. 1728 ii. \(\frac{1}{512}\)iii. 0.000169
Answer» i. 1728= 12 x 12 x 12 1728= 123 ii. \(\frac{1}{512} = \frac{1}{2^9}\) = 2^-9 iii. 0.000169 = (0.013)^-2

90.

Express the following numbers in the exponential form.i. 1728 ii. \(\frac{1}{512}\)iii. 0.000169

Answer»

i. 1728= 12 x 12 x 12

1728= 123

ii. \(\frac{1}{512} = \frac{1}{2^9}\) = 2^-9

iii. 0.000169 = (0.013)^-2

Discussion

91.	(22)3 = ………………A) 512 B) 64C) 223D) 256
Answer» Correct option is B) 64 (2²)³ = 2^{2 x 3} = 2⁶ = 64

91.

(22)3 = ………………A) 512 B) 64C) 223D) 256

Answer»

Correct option is B) 64

(2²)³ = 2^{2 x 3} = 2⁶ = 64

Discussion

92.	Which is larger (53 x 54 x 55 x 56) or (57 x 58)
Answer» 5³ x 5⁴ x 5⁵ x 5⁶ = 5^3+4+5+6 = 5¹⁸x 5⁷ x 5⁸ = 5⁷⁺⁸ = 5⁸ ∴ 5³ x 5⁴ x 5⁵x 5⁶ is larger than 5⁷ x 5⁸

92.

Which is larger (53 x 54 x 55 x 56) or (57 x 58)

Answer»

5³ x 5⁴ x 5⁵ x 5⁶

= 5^3+4+5+6

= 5¹⁸x 5⁷ x 5⁸

= 5⁷⁺⁸

= 5⁸

∴ 5³ x 5⁴ x 5⁵x 5⁶ is larger than 5⁷ x 5⁸

Discussion

93.	Which of the following is not true?A) x-7 = \(\frac{1}{x^7}\)B) x5 = \(\frac{1}{x^-5}\)C) x° = 1 D) \(\frac{1}{x^-5}\) = x1/5
Answer» Correct option is D) \(\frac{1}{x^-5}\) = x^1/5 ^{\(\frac1{x^{-5}}=(x^{-5})^{-1}\)}= x⁵ \(\neq\) x^1/5 correct answer is D because 1/x^-5= x^5

93.

Which of the following is not true?A) x-7 = \(\frac{1}{x^7}\)B) x5 = \(\frac{1}{x^-5}\)C) x° = 1 D) \(\frac{1}{x^-5}\) = x1/5

Answer»

Correct option is D) \(\frac{1}{x^-5}\) = x^1/5

^{\(\frac1{x^{-5}}=(x^{-5})^{-1}\)}= x⁵ \(\neq\) x^1/5

correct answer is D

because

1/x^-5= x^5

Discussion

94.	Find the value of (-0.2)-4
Answer» (-0.2)^-4= (-2/10)^-4 = (-1/5)^-4 = (-5)⁴ = -5 x -5 x - 5 x -5 = 625

94.

Find the value of (-0.2)-4

Answer»

(-0.2)^-4= (-2/10)^-4

= (-1/5)^-4

= (-5)⁴

= -5 x -5 x - 5 x -5

= 625

Discussion

95.	How many zeros are there in 104 x 103 x 102 x 10?
Answer» 10⁴ × 10³ × 10² × 10 = 10^4+3+2+1 = 10¹⁰ There are 10 zeros.

95.

How many zeros are there in 104 x 103 x 102 x 10?

Answer»

10⁴ × 10³ × 10² × 10

= 10^4+3+2+1

= 10¹⁰

There are 10 zeros.

Discussion

96.	Express 5-3 as a rational number.
Answer» 5^-3 We know that, = (5)^-3 = (1/5)³ … [∵ (a/b)^-n = (b/a)ⁿ] = (1³/5³) = (1/125)

96.

Express 5-3 as a rational number.

Answer»

5^-3

We know that,

= (5)^-3 = (1/5)³ … [∵ (a/b)^-n = (b/a)ⁿ]

= (1³/5³)

= (1/125)

Discussion

97.	Which of the following is true? A) 138/135 = 1313B) (2 × 3)4 = 34/24 C) (32)3 = 36 D) ( 3/4)5 = 35 × 34
Answer» Correct option is C) (3²)³ = 3⁶ (A) \(\frac{13^8}{13^5}=13^{8-5}=13^3\neq13^{13}\) (B) (2 x 3)⁴ = 2⁴ x 3⁴ \(\neq\) \(\frac{3^4}{2^4}\) (C) (3²)³ = 3^{2 x 3} = 3⁶ (D) (3/4)⁵ = \(\frac{3^5}{4^5}\neq3^5\times3^4\)

97.

Which of the following is true? A) 138/135 = 1313B) (2 × 3)4 = 34/24 C) (32)3 = 36 D) ( 3/4)5 = 35 × 34

Answer»

Correct option is C) (3²)³ = 3⁶

(A) \(\frac{13^8}{13^5}=13^{8-5}=13^3\neq13^{13}\)

(B) (2 x 3)⁴ = 2⁴ x 3⁴ \(\neq\) \(\frac{3^4}{2^4}\)

(C) (3²)³ = 3^{2 x 3} = 3⁶

(D) (3/4)⁵ = \(\frac{3^5}{4^5}\neq3^5\times3^4\)

Discussion

98.	Express (-13/11)2 a rational number.
Answer» (-13/11)² We have, (-13/11)² = (-13²/11²) = (-13 × -13) / (11 × 11) = (169/121)

98.

Express (-13/11)2 a rational number.

Answer»

(-13/11)²

We have,

(-13/11)² = (-13²/11²)

= (-13 × -13) / (11 × 11)

= (169/121)

Discussion

99.	Simplify:(i) (3/4)2(ii) (-2/3)4(iii) (-4/5)5
Answer» (i) Given (3/4)² (3/4)² = (3/4) × (3/4) = (9/16) (ii) Given (-2/3)⁴ (-2/3)⁴ = (-2/3) × (-2/3) × (-2/3) × (-2/3) = (16/81) (iii) Given (-4/5)⁵ (-4/5)⁵ = (-4/5) × (-4/5) × (-4/5) × (-4/5) × (-4/5) = (-1024/3125)

99.

Simplify:(i) (3/4)2(ii) (-2/3)4(iii) (-4/5)5

Answer»

(i) Given (3/4)²

(3/4)²

= (3/4) × (3/4)

= (9/16)

(ii) Given (-2/3)⁴

(-2/3)⁴

= (-2/3) × (-2/3) × (-2/3) × (-2/3)

= (16/81)

(iii) Given (-4/5)⁵

(-4/5)⁵

= (-4/5) × (-4/5) × (-4/5) × (-4/5) × (-4/5)

= (-1024/3125)

Discussion

100.	Simplify:(i) (-2) × (-3)3(ii) (-3)2 × (-5)3(iii) (-2)5 × (-10)2
Answer» (i) Given (-2) × (-3)³ (-2) × (-3)³ = (-2) × (-3) × (-3) × (-3) = (-2) × (-27) = 54 (ii) Given (-3)² × (-5)³ (-3)² × (-5)³ = (-3) × (-3) × (-5) × (-5) × (-5) = 9 × (-125) = -1125 (iii) Given (-2)⁵ × (-10)² (-2)⁵ × (-10)² = (-2) × (-2) × (-2) × (-2) × (-2) × (-10) × (-10) = (-32) × 100 = -3200

100.

Simplify:(i) (-2) × (-3)3(ii) (-3)2 × (-5)3(iii) (-2)5 × (-10)2

Answer»

(i) Given (-2) × (-3)³

(-2) × (-3)³

= (-2) × (-3) × (-3) × (-3)

= (-2) × (-27)

= 54

(ii) Given (-3)² × (-5)³

(-3)² × (-5)³

= (-3) × (-3) × (-5) × (-5) × (-5)

= 9 × (-125)

= -1125

(iii) Given (-2)⁵ × (-10)²

(-2)⁵ × (-10)²

= (-2) × (-2) × (-2) × (-2) × (-2) × (-10) × (-10)

= (-32) × 100

= -3200

Discussion

Explore topic-wise InterviewSolutions in .

If 64a = \(\frac{1}{256^b}\), then 3a+4b equals(a) 2 (b) 4 (c) 8 (d) 0

(1/2)-2 + (1/3)-2 + (1/4)-2 = ?(a) (61/144) (b) 29 (c) (144/61) (d) none of these

43.5 : 25 is the same as(a) 4 : 1 (b) 2 : 1 (c) 7 : 5 (d) 7 : 10

{6-1+(3/2)-1}-1(a) (2/3) (b) (5/6) (c) (6/5) (d) None of these

{(3/4)-1 – (1/4)-1}-1 = ?(a) (3/8) (b) (-3/8) (c) (8/3) (d) (-8/3)

(-1/2)-6 = ?(a) -64 (b) 64 (c) (1/64) (d) (-1/64)

(2/3)0 = ?A. 3/2B. 2/3C. 1D. 0

(-5/3)-1 = 1A. 5/3B. 3/5C. -3/5D. None of these

(-1/2)3 = ?A. -1/6B. 1/6C. 1/8D. -1/8

(-3/4)2 = ?A. -9/16B. 9/16C. 16/9D. -16/9

[{(-1/2)2}-2]-1=?(a) (1/16) (b) 16 (c) (-1/16) (d) -16

Find the reciprocal of (-5/6)11.

What is 10-10 equal to?

The stand form of the distance between the earth and the moon is aprox. 12,000,000,000 mts A) 12 × 1011 years B) 1.2 × 1010 years C) 0.12 × 109 years D) 120 × 1011 years

Prime factorisation of 72 is ………….?A) 23 × 32B) 23 × 33C) 33 × 22 D) 2 × 32 × 7

(1/2)-2 + (1/3)-2 + (1/4)-2 = ?A. 61/144B. 144/61C. 29D. 1/29

Express each of the following numbers as a product of powers of their prime factors:(i) 450(ii) 2800(iii) 24000

r × r × 7 times = ………………? A) 7rB) 7r C) r7D) None

Express (5/7)-1× (7/4)-1 as a rational number.

Express (-6)-1 as a rational number.

If x = yz , y = zx and z = xy , then(a) \(\frac{xy}{z} = 1\)(b) xyz = 1 (c) x + y + z = 1 (d) xz = y

Find the value of x if \(\big[3^{2x-2}+10\big]\) ÷ 13 = 7.(a) 1 (b) 3 (c) 4 (d) 2

Express (-2)-5 as a rational number.

Express each of the following numbers as a product of powers of their prime factors:(i) 36(ii) 675(iii) 392

By what number should three be multiplied so that the product is 729’?

Express each of the following numbers in exponential form:(i) 512(ii) 625(iii) 729

If a = 3, b = 2 find the value of(i) ab + ba(ii) aa + bb(iii) (a + b)ab(iv)(a – b)a

Express each of the following in exponential form:(i) x × x × x × x × a × a × b × b × b(ii) (-2) × (-2) × (-2) × (-2) × a × a × a(iii) (-2/3) × (-2/3) × x × x × x

Identify the greater number in each of the following pairs.(i) 23 or 32(ii) 53 or 35(iii) 28 or 82

Express each of the following in exponential form:(i) (-5) × (-5) × (-5)(ii) (-5/7) × (-5/7) × (-5/7) × (-5/7)(iii) (4/3) × (4/3) × (4/3) × (4/3) × (4/3)

Identify the greater number in each of the following:(i) 25 or 52(ii) 34 or 43(iii) 35 or 53

Express (4)-1 as a rational number.

Express (-1)9 a rational number.

Express (-1/2)5 a rational number.

Express (2/3)5 a rational number.

Express (-4/7)3 a rational number.

Express (-8/5)3 a rational number.

Express (1/6)3 a rational number.

Express the following numbers in the exponential form.i. 1728 ii. \(\frac{1}{512}\)iii. 0.000169

(22)3 = ………………A) 512 B) 64C) 223D) 256

Which is larger (53 x 54 x 55 x 56) or (57 x 58)

Which of the following is not true?A) x-7 = \(\frac{1}{x^7}\)B) x5 = \(\frac{1}{x^-5}\)C) x° = 1 D) \(\frac{1}{x^-5}\) = x1/5

Find the value of (-0.2)-4

How many zeros are there in 104 x 103 x 102 x 10?

Express 5-3 as a rational number.

Which of the following is true? A) 138/135 = 1313B) (2 × 3)4 = 34/24 C) (32)3 = 36 D) ( 3/4)5 = 35 × 34

Express (-13/11)2 a rational number.

Simplify:(i) (3/4)2(ii) (-2/3)4(iii) (-4/5)5

Simplify:(i) (-2) × (-3)3(ii) (-3)2 × (-5)3(iii) (-2)5 × (-10)2