Exponents And Powers Class 8th Mathematics AP Board Solution

Class 8th Mathematics AP Board Solution
Exercise 4.1
  1. 4−3 Simplify and give reasons
  2. (−2)^7 Simplify and give reasons
  3. Simplify and give reasons
  4. (−3)−4 Simplify and give reasons
  5. (−2)^7 × (−2)^3 × (−2)^4 Simplify the following:
  6. (1/2)^4 x (1/2)^5 x (1/2)^6 Simplify the following:
  7. Simplify the following: 4^4 (5/4)^4
  8. (5^-4/5^-6) ×5^3 Simplify the following:
  9. (−3)^4 × 7^4 Simplify the following:
  10. 2^2 × 3^2/2^-2 ×3-1 Simplify
  11. (4−1 × 3−1) ÷ 6−1 Simplify
  12. (4^0 + 5−1) ×5^2 × 1/3 Simplify and give reasons
  13. (1/2)^-3 x (1/4)^-3 x (1/5)^-3 Simplify and give reasons
  14. (2−1 + 3−1 + 4−1) × 3/4 Simplify and give reasons
  15. 3^-2/3 ×(3^0 -3-1) Simplify and give reasons
  16. 1 + 2−1 + 3−1 + 4^0 Simplify and give reasons
  17. [(3/2)^-2]^2 Simplify and give reasons
  18. [(3^2 - 2^2) / 1/5]^2 Simplify and give reasons
  19. ((5^2)^3 × 5^4) ÷ 5^6 Simplify and give reasons
  20. (2/3)^3 x (2/3)^5 = (2/3)^n-2 Find the value of ‘n’ in each of the following:…
  21. (−3)n+1 × (−3)^5 = (−3)−4 Find the value of ‘n’ in each of the following:…
  22. 72n+1 ÷ 49 = 7^3 Find the value of ‘n’ in each of the following:
  23. Find ‘x’ if 2^-3 = 1/2^x
  24. Simplify [(3/4)^-2 / (4/5)^-3] x (3/5)^-2
  25. 9m^2 - 10n^3 If m = 3 and n = 2 find the value of
  26. 2m^2 n^2 If m = 3 and n = 2 find the value of
  27. 2m^3 + 3n^2 - 5m^2 n If m = 3 and n = 2 find the value of
  28. mn - nm If m = 3 and n = 2 find the value of
  29. Simplify and give reasons (4/7)^-1 x (7/4)^-7
Exercise 4.2
  1. 0.000000000947 Express the following numbers in the standard form.…
  2. 543000000000 Express the following numbers in the standard form.
  3. 48300000 Express the following numbers in the standard form.
  4. 0.00009298 Express the following numbers in the standard form.
  5. 0.0000529 Express the following numbers in the standard form.
  6. 4.37 × 10^5 Express the following numbers in the usual form.
  7. 5.8 ×10^7 Express the following numbers in the usual form.
  8. 32.5 × 10−4 Express the following numbers in the usual form.
  9. 3.71529 × 10^7 Express the following numbers in the usual form.
  10. 3789 × 10−5 Express the following numbers in the usual form.
  11. 24.36 × 10−3 Express the following numbers in the usual form.
  12. Size of the bacteria is 0.0000004 m Express the following information in the…
  13. The size of red blood cells is 0.000007mm Express the following information in…
  14. The speed of light is 300000000 m/sec Express the following information in the…
  15. The distance between the moon and the earth is 384467000 m (app) Express the…
  16. The charge of an electron is 0.0000000000000000016 coulombs Express the…
  17. Thickness of a piece of paper is 0.0016 cm Express the following information in…
  18. The diameter of a wire on a computer chip is 0.000005 cm Express the following…
  19. In a pack, there are 5 books, each of thickness 20 mm and 5 paper sheets each of…
  20. x-3 × x-2 = x-6 Rakesh solved some problems of exponents in the following way.…
  21. (x^2)^3 = 3x^-1 = 1/3x root x = x^8/x^2 = x^4 Rakesh solved some problems of…
  22. (x^2)^3 = x^2^3 = x^6 Rakesh solved some problems of exponents in the following…
  23. x^-2 = root x Rakesh solved some problems of exponents in the following way. Do…
  24. 3x^-1 = 1/3x Rakesh solved some problems of exponents in the following way. Do…

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


7128



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: am × 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


=


=


= 56-1


= 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


= 252-1


= 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)


= 56+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 =  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


72n+1-2 = 73


72n– 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 – nm


Answer:

Given, = mn – nm

= 32 23


= 9  8


= 1


 mn – nm = 1



Question 29.

Simplify and give reasons


Answer:

Given, = 




= ()2


 = ()2




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 = x2x3


= 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


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