Exponents And Powers Solution of TS & AP Board Class 8 Mathematics

Exponents And Powers Solution of TS & AP Board Class 8 Mathematics

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Exercise 4.1

Question 1.

Simplify and give reasons

4⁻³

Answer:

Given, = 

= 

= 

= 

Question 2.

Simplify and give reasons

(−2)⁷

Answer:

Given, = (2)⁷

= -(2×2×2×2×2×2×2 )∵(-2)⁷=-(2)⁷

= 128

⁷= 128

Question 3.

Simplify and give reasons

Answer:

Given, =

= 

= 

=

Question 4.

Simplify and give reasons

(−3)⁻⁴

Answer:

Given, =(3)^-4

= 

= 

= 

 (3)^-4= 

Question 5.

Simplify the following:

(−2)⁷ × (−2)³ × (−2)⁴

Answer:

Given, = (−2)⁷ × (−2)³ × (−2)⁴

= (−2)⁷⁺³⁺⁴ 

= (−2)¹⁴

(−2)⁷ × (−2)³ × (−2)⁴=(−2)¹⁴

Question 6.

Simplify the following:

Answer:

To Find: 

We know that: a^m × aⁿ = a(m+n)
Therefore,

=2^-15

Answer.

Question 7.

Simplify the following:
4⁴ ×

Answer:

Given, = 4⁴ ×

= 4⁴

= 5⁴
= 625
Question 8.

Simplify the following:

×5³

Answer:

Given, = ×5³

= ³

=

=

= 5^6-1

= 5⁵

= 625

×5³ = 625.

Question 9.

Simplify the following:

(−3)⁴ × 7⁴

Answer:

Given, = (−3)⁴ × 7⁴

= 

= 992401

= 194,481

(−3)⁴ × 7⁴= 194,481

Question 10.

Simplify

2²× ×3^-1

Answer:

Given, =2²× ×3^-1

=  ( )

=  ( )

= 2⁴ × 3¹

= 16 × 3

= 48

2²× ×3^-1 = 48.

Question 11.

Simplify

(4⁻¹ × 3⁻¹) ÷ 6⁻¹

Answer:

Given, = (4⁻¹ × 3⁻¹) ÷ 6⁻¹

= 

= 

= 

= 

(4⁻¹ × 3⁻¹) ÷ 6⁻¹ = 

Question 12.

Simplify and give reasons

(4⁰ + 5⁻¹) ×5² ×

Answer:

Given, =(4⁰ + 5⁻¹) ×5² ×

=(1+)5²

= 5²

= 25^2-1

= 25

= 10

 (4⁰ + 5⁻¹) ×5² × = 10

Question 13.

Simplify and give reasons

Answer:

Given, = 

= 

= 2³4³5³

= 2³(2²)³5³

= 2³2⁶5³

= 2³⁺⁶5³

= 2⁹5³

 = 2⁹5³

Question 14.

Simplify and give reasons

(2⁻¹ + 3⁻¹ + 4⁻¹) ×

Answer:

Given, = (2⁻¹ + 3⁻¹ + 4⁻¹) ×

= ()

= ()

= 

= 

= 

(2⁻¹ + 3⁻¹ + 4⁻¹) × = 

Question 15.

Simplify and give reasons

×(3⁰ -3^-1)

Answer:

Given, = ×(3⁰ -3^-1)

= )

= 

= 

= 

= 

×(3⁰ -3^-1) = 

Question 16.

Simplify and give reasons

1 + 2⁻¹ + 3⁻¹ + 4⁰

Answer:

Given, = 1 + 2⁻¹ + 3⁻¹ + 4⁰

= 1 +  +  + 1 

= 

= 

1 + 2⁻¹ + 3⁻¹ + 4⁰ = 

Question 17.

Simplify and give reasons

Answer:

Given, = 

= []²

= 

= 

= 

Question 18.

Simplify and give reasons

Answer:

Given, = 

= [(9-4)5]²

= [55]²

= [5²]²

= 5⁴

=625

= 625

Question 19.

Simplify and give reasons

((5²)³ × 5⁴) ÷ 5⁶

Answer:

Given, = ((5²)³ × 5⁴) ÷ 5⁶

= (5⁶5⁴)

= (5⁶5⁴5^-6)

= 5^6+4-6

= 5⁴

= 625

((5²)³ × 5⁴) ÷ 5⁶ = 625

Question 20.

Find the value of ‘n’ in each of the following:

Answer:

Given, 

 = (

 = (

 = (

 Comparing powers of  , we get

 n2=8

 n=8+2

 n=10

Question 21.

Find the value of ‘n’ in each of the following:

(−3)ⁿ⁺¹ × (−3)⁵ = (−3)⁻⁴

Answer:

Given, (−3)ⁿ⁺¹ × (−3)⁵ = (−3)⁻⁴

(3)ⁿ⁺¹⁺⁵ = (−3)⁻⁴

(3)ⁿ⁺⁶ = (−3)⁻⁴

 Comparing powers of (3), we get

 n+6 = 4

 n = 4  6

 n=  10

Question 22.

Find the value of ‘n’ in each of the following:

7²ⁿ⁺¹ ÷ 49 = 7³

Answer:

Given, 7²ⁿ⁺¹ ÷ 49 = 7³

7²ⁿ⁺¹ = 7³

7²ⁿ⁺¹ = 7³

7²ⁿ⁺¹ = 7³

7^2n+1-2 = 7³

7^{2n– 1} = 7³

 By comparing powers of 7, we get

 2n1 = 3

 2n = 3+1

 2n = 4

 n = 

 n = 2

Question 23.

Find ‘x’ if 

Answer:

Given, 

Then, find the value of x.

 = 

 By comparing the powers of 2, we get

x = 3

 x = 3

Question 24.

Simplify

Answer:

Given, = 

= [] 

= 

= 

= 

= 

 = 

Question 25.

If m = 3 and n = 2 find the value of

9m² – 10n³

Answer:

Given, = 9m² – 10n³

Where, m = 3 and n = 2

= 93²102³

= (99)  (108)

= 81  80

= 1

 9m² – 10n³ = 1

Question 26.

If m = 3 and n = 2 find the value of

2m² n²

Answer:

Given, = 2m² n²

 m = 3 and n = 2

 2m² n² = 23²2²

= 294

= 72

 2m² n² = 72

Question 27.

If m = 3 and n = 2 find the value of

2m³ + 3n² – 5m²n

Answer:

Given, = 2m³ + 3n² – 5m²n

= 23³ + 32² 53²2

= 54 + 12  90

=  24

 2m³ + 3n² – 5m²n =  24

Question 28.

If m = 3 and n = 2 find the value of

mⁿ – n^m

Answer:

Given, = mⁿ – n^m

= 3² 2³

= 9  8

= 1

 mⁿ – n^m = 1

Question 29.

Simplify and give reasons

Answer:

Given, = 

= 

= 

= 

= ()²

 = ()²

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Exercise 4.2

Question 1.

Express the following numbers in the standard form.

0.000000000947

Answer:

Given, = 0.000000000947

= 0.000000000947

= 9.47 

= 9.47 

 0.000000000947 = 9.47 

Question 2.

Express the following numbers in the standard form.

543000000000

Answer:

Given, = 543000000000

= 543000000000 

= 5.43  10¹¹

 9.47  = 5.43  10¹¹

Question 3.

Express the following numbers in the standard form.

48300000

Answer:

Given, = 48300000

= 48300000 

= 4.83  10⁷

 48300000 = 4.83  10⁷

Question 4.

Express the following numbers in the standard form.

0.00009298

Answer:

Given, = 0.00009298

= 0.00009298 

= 9.28 

= 9.28 

 0.00009298 = 9.28 

Question 5.

Express the following numbers in the standard form.

0.0000529

Answer:

Given, = 0.0000529

= 0.0000529 

= 5.29 

 0.0000529 = 5.29 

Question 6.

Express the following numbers in the usual form.

4.37 × 10⁵

Answer:

Given, = 4.37 × 10⁵

= 4.37  100000

= 437000

4.37 × 10⁵ = 437000.

Question 7.

Express the following numbers in the usual form.

5.8 ×10⁷

Answer:

Given, = 5.8 ×10⁷

= 5.8  10000000

= 58000000

5.8 ×10⁷ = 58000000.

Question 8.

Express the following numbers in the usual form.

32.5 × 10⁻⁴

Answer:

Given, = 32.5 × 10⁻⁴

= 32.5 

= 0.00325

32.5 × 10⁻⁴ = 0.00325

Question 9.

Express the following numbers in the usual form.

3.71529 × 10⁷

Answer:

Given, = 3.71529 × 10⁷

= 3.71529  10000000

= 37152900

3.71529 × 10⁷ = 37152900

Question 10.

Express the following numbers in the usual form.

3789 × 10⁻⁵

Answer:

Given, = 3789 × 10⁻⁵

= 3789 

= 0.03789

3789 × 10⁻⁵ = 0.03789

Question 11.

Express the following numbers in the usual form.

24.36 × 10⁻³

Answer:

Given, = 24.36 × 10⁻³

= 24.36 

= 0.02436

24.36 × 10⁻³ = 0.02436

Question 12.

Express the following information in the standard form

Size of the bacteria is 0.0000004 m

Answer:

Given, Size of the bacteria is = 0.0000004 m

= 0.0000004 

= 4

= 4 

 Size of the bacteria is = 4 m

Question 13.

Express the following information in the standard form

The size of red blood cells is 0.000007mm

Answer:

Given, The size of red blood cells is = 0.000007mm

= 0.000007 

= 7 

The size of red blood cells = 7 mm

Question 14.

Express the following information in the standard form

The speed of light is 300000000 m/sec

Answer:

Given, The speed of light is = 300000000 m/sec

= 300000000 

= 3.0  10⁸

 The speed of light = 3.0  10⁸m/sec.

Question 15.

Express the following information in the standard form

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

Answer:

Given, The distance between the moon and the earth,

= 384467000 

= 3.84467 10⁸

 The distance between the moon and the earth=3.84467 10⁸ m

Question 16.

Express the following information in the standard form

The charge of an electron is 0.0000000000000000016 coulombs

Answer:

Given, The charge of an electron = 0.0000000000000000016 coulombs.

= 0.0000000000000000016 

= 1.6 

 The charge of an electron = 1.6  coulombs.

Question 17.

Express the following information in the standard form

Thickness of a piece of paper is 0.0016 cm

Answer:

Given, Thickness of a piece of paper = 0.0016 cm

= 0.0016 

= 1.6 

= 1.6 

 Thickness of a piece of paper = 1.6 cm.

Question 18.

Express the following information in the standard form

The diameter of a wire on a computer chip is 0.000005 cm

Answer:

Given, diameter of a wire on a computer chip = 0.000005 cm.

= 0.000005 

= 5 

= 5 

 The diameter of a wire on a computer chip = 5 cm.

Question 19.

In a pack, there are 5 books, each of thickness 20 mm and 5 paper sheets each of thickness 0.016mm. What is the total thickness of the pack.

Answer:

Given, thickness of book = 20mm

Thickness of paper = 0.016mm

The total thickness of the pack is to be found.

 the total thickness of the pack = (520) + (50.016)

= 100 + 0.08

= 100.08

= 1.0008  10²mm

 the total thickness of the pack = 1.0008  10²mm

Question 20.

Rakesh solved some problems of exponents in the following way. Do you agree with the solutions? If not why? Justify your argument.

x^–3 × x^–2 = x^–6

Answer:

Given, x^–3 × x^–2 = x^–6

 the solution is wrong, because

 x^–3 × x^–2 = x^-3-2

= x^-5

 x^–3 × x^–2 x^–6

Question 21.

Rakesh solved some problems of exponents in the following way. Do you agree with the solutions? If not why? Justify your argument.

Answer:

Given, 

 = 

= 

Question 22.

Rakesh solved some problems of exponents in the following way. Do you agree with the solutions? If not why? Justify your argument.

Answer:

Given, 

 (x²)³ = x^2x3

= x⁶

 the solution by Rakesh is wrong.

Question 23.

Rakesh solved some problems of exponents in the following way. Do you agree with the solutions? If not why? Justify your argument.

Answer:

Given, 

 = 

Question 24.

Rakesh solved some problems of exponents in the following way. Do you agree with the solutions? If not why? Justify your argument.

Answer:

Given, 

 3 = 3

= 

 3