Exponents and Powers Worksheet

Multiple Choice Questions

1. Find the value of x in the following
    
   (a) -2
   (b) -3
   (c) 2
   (d) 3
   
   
2. Diameter of a unnamed planet is 2000 million km, which of the following represents the radius of this planet
   (a) 2 × 10⁶ km
   (b) 1 × 10⁶ km
   (c) 1 × 10⁹ km
   (d) 2 × 10⁹ km
   
   
3.  
   (a) 
   (b) 
   (c) 
   (d) 
   
   
4. One micrometre is one millionth of a metre, which of the following represents 1 micrometre
   (a) 1 × 10⁶ m
   (b) 1 × 10^-6 m
   (c) 1 × 10⁵ m
   (d) 1 × 10^-5 m
   
   
5. Which of the following represents the product of 1.41 × 0.02 × 0.0001
   (a) 2.82 × 10^-8
   (b) 2.82 × 10^-7
   (c) 2.82 × 10^-6
   (d) 2.82 × 10^-5
   
   
6. Find the value of ‘t’ for which 
   (a) -7
   (b) 7
   (c) -5
   (d) 5
   
   
7. The product of  ten-thousandth and hundredths is
   (a) 1 × 10^-6
   (b) 1 × 10^-5
   (c) 1 × 10^-4
   (d) 1 × 10^-3
   
   
8. Two-thousandth is divided by four-hundredth and the division is multiplied by eight-tenths. So the result will be
   (a) 2 × 10^-2
   (b) 4 × 10^-2
   (c) 2 × 10^-4
   (d) 4 × 10^-4
   
   
9. A pile of paper is stacked one over another, now if the total thickness of the pile is 20 and the thickness of each piece of paper is 0.0016 cm, then find the total number of papers in the pile
   (a) 125 pages
   (b) 1250 pages
   (c) 12500 pages
   (d) 125000 pages
   
   
10. Express the inverse of 4^th power of 16 as a power with the base 2
    (a) 2¹⁶
    (b) 2⁸
    (c) 2^-8
    (d) 2^-16
    
    Subjective Questions
    
11. A stick of length 4² cm is broken at the end by 4 cm. Now if the larger part is further divided into two halves, then find the length of each half.
    
    
12. The average diameter of human hair is 0.00002 m, now find the average radius of human hair and express it in exponential form.
    
    
13. Represent the following exponential form
    There are three water storage tanks of ‘x’ litre each, first one is filled half with water, second one is filled one-fourth, and third one is filled one-eight. Find the total amount of water stored in all three tanks combined.
    
    
14. Divide one millionth by one thousandth, then multiply the division by one hundredth.
    
    
15. The cost of 1 gram of iron is Rs. 0.07, if Ram needs 2.345 grams of iron for his experiment then find the total cost of iron that needs to be purchased.
    
    
16. A big container having a capacity of 5.5 litres is filled with kerosene to its full capacity. Now this kerosene it to be filled into smaller containers of capacity 0.11 litres each. So find the total number of small containers can be filled with the available kerosene.
    
    
17. The radius of an isolated atom is 300 picometre. Also it is found that the radius of an atom is 10,000 times the radius of its nucleus. If 1 picometre = 1 × 10⁻¹² m, then find the radius of the nucleus.
    
    
18. The radius of a planet Mandalore is 6800000m and that of Naboo is 1700m. Now compare the radius of two planets and express the radius of Naboo in terms of radius of Mandalore.
    
    
19. Rajeev’s monthly income is 4⁸ rupees, he spends 2¹⁰ for monthly electric bill and 215 for other expenses. Rest of the money he saves. Now find the ratio of Rajeev’s savings to his expenses.
    
    
20. One mole has 6. 022 × 10²³ atoms. Now we take 2.4 moles of A and 1.05 moles of B and mix them together, then find the total number of atoms in the mixture.

Multiple Choice Questions

1. Find the value of x in the following
    
   (a) -2
   (b) -3
   (c) 2
   (d) 3
   
   Correct option: (b) -3
   Solution:
   
   ∴ 2^-x = 9 - 1
   ∴ 2^-x = 9
   ∴ 2^-x = 2³
   ∴ x = -3
   
   
2. Diameter of a unnamed planet is 2000 million km, which of the following represents the radius of this planet
   (a) 2 × 10⁶ km
   (b) 1 × 10⁶ km
   (c) 1 × 10⁹ km
   (d) 2 × 10⁹ km
   
   Correct option: (c) 1 × 10⁹ km
   Solution: 
   2000 million = 2000000000
   ∴ Diameter = 2 × 10⁹ km
   ∴ Radius = 1 × 10⁹ km
   
   
3.  
   (a) 
   (b) 
   (c) 
   (d) 
   
   Correct option: (a) 
   Solution: 
   
   
   
4. One micrometre is one millionth of a metre, which of the following represents 1 micrometre
   (a) 1 × 10⁶ m
   (b) 1 × 10^-6 m
   (c) 1 × 10⁵ m
   (d) 1 × 10^-5 m
   
   Correct option: (b) 1 × 10^-6
   Solution:
   One millionth of a metre =  metre = 1 × 10^-6m
   
   
5. Which of the following represents the product of 1.41 × 0.02 × 0.0001
   (a) 2.82 × 10^-8
   (b) 2.82 × 10^-7
   (c) 2.82 × 10^-6
   (d) 2.82 × 10^-5
   
   Correct option: (c) 2.82 × 10^-6
   Solution:
   1.41 × 0.02 × 0.0001
   = 141 × 10^-2 × 2 × 10^-2 × 1 × 10^-4
   = 282 × 10^-8
   = 2.82 × 10^-6
   
   
6. Find the value of ‘t’ for which 
   (a) -7
   (b) 7
   (c) -5
   (d) 5
   
   Correct option: (b) 7
   Solution:
   
   ∴ 3^-9 x 3^2t = 3⁵
   ∴ 3^{-9 + 2t} = 3⁵
   ∴ -9 + 2t = 5
   ∴ 2t = 14
   ∴ t = 7
   
   
7. The product of  ten-thousandth and hundredths is
   (a) 1 × 10^-6
   (b) 1 × 10^-5
   (c) 1 × 10^-4
   (d) 1 × 10^-3
   
   Correct option: (a) 1 × 10^-6
   Solution:
   ten-thousandth = 
   hunderdth = 
   Hence,
   1 x 10^-4 x 1 x 10^-2
   = 1 x 10^-4-2
   = 1 x 10^-6
   
   
8. Two-thousandth is divided by four-hundredth and the division is multiplied by eight-tenths. So the result will be
   (a) 2 × 10^-2
   (b) 4 × 10^-2
   (c) 2 × 10^-4
   (d) 4 × 10^-4
   
   Correct option: (b) 4 × 10^-2
   Solution:
   
   
   
9. A pile of paper is stacked one over another, now if the total thickness of the pile is 20 and the thickness of each piece of paper is 0.0016 cm, then find the total number of papers in the pile
   (a) 125 pages
   (b) 1250 pages
   (c) 12500 pages
   (d) 125000 pages
   
   Correct option: (c) 12500 pages
   Solution:
   Thickness of pile = 20 cm = 2 x 10¹ cm
   Thickness of paper = 0.0016 cm 16 x 10^-4
   Total number of papers = 
   
   = 0.125 x 10⁵
   = 12500 pages
   
   
10. Express the inverse of 4^th power of 16 as a power with the base 2
    (a) 2¹⁶
    (b) 2⁸
    (c) 2^-8
    (d) 2^-16
    
    Correct option: (d) 2^-16
    Solution:
    4th power of 16 = 16⁴
    Inverse = 16^-4
    ∴ (16)^-4
    = (2⁴)^-4
    = 2^-16
    
    Subjective Questions
    
    
11. A stick of length 4² cm is broken at the end by 4 cm. Now if the larger part is further divided into two halves, then find the length of each half.
    
    Solution:
    Lenght of larger part = 4² - 4
    Half of larger part = 
    
    = 2⁴ x 2^-1 - 2² x 2^-1
    = 2³ -2¹
    = 8 - 2
    = 6
    
    
12. The average diameter of human hair is 0.00002 m, now find the average radius of human hair and express it in exponential form.
    
    Solution:
    Avg. diameter of hair = 0.00002m = 2 x 10^-5m
    ∴ Avg. radius of hair =  = 1 x 10^-5m
    
    
13. Represent the following exponential form
    There are three water storage tanks of ‘x’ litre each, first one is filled half with water, second one is filled one-fourth, and third one is filled one-eight. Find the total amount of water stored in all three tanks combined.
    
    Solution:
    
    = 2^-1 x x + 2^-2 x + 2^-3 x x
    = (2^-1 + 2^-2 + 2^-3)x
    Note: This example can be also attempted using addition of fractions.
    
    
14. Divide one millionth by one thousandth, then multiply the division by one hundredth.
    
    Solution:
    
    
    
15. The cost of 1 gram of iron is Rs. 0.07, if Ram needs 2.345 grams of iron for his experiment then find the total cost of iron that needs to be purchased.
    
    Solution:
    Cost of 1 gram (iron) = Rs. 0.07 = 7 × 10^-2
    Hence, cost of 2.345 grams (iron) = 7 × 10^-2 × 2345 × 10^-3
    =16415 × 10^-5
    = Rs. 16.415 × 10^-2
    
    
16. A big container having a capacity of 5.5 litres is filled with kerosene to its full capacity. Now this kerosene it to be filled into smaller containers of capacity 0.11 litres each. So find the total number of small containers can be filled with the available kerosene.
    
    Solution:
    Capacity of big container = 5.5 litres
    Capacity of small container = 0.11 litres
    No.of smaller containers = 
    
    = 5 x 101
    = 50
    
    
17. The radius of an isolated atom is 300 picometre. Also it is found that the radius of an atom is 10,000 times the radius of its nucleus. If 1 picometre = 1 × 10⁻¹² m, then find the radius of the nucleus.
    
    Solution:
    Radius of atom = 300 x 10^-12 m
    Radius of nucleus = 
    
    = 300 x 10^-12 x 10^-4
    = 300 x 10^-16
    = 3 x 10^-14
    
    
18. The radius of a planet Mandalore is 6800000m and that of Naboo is 1700m. Now compare the radius of two planets and express the radius of Naboo in terms of radius of Mandalore.
    
    Solution:
    Radius of Mandalore = 6.8 x 10⁶m
    Radius of Naboo = 1.7 x 10³m
    
    = 0.25 × 10^-3
    ∴ Radius of Naboo - 2.5 x 10^-2 x Radius of Mandalore
    
    
19. Rajeev’s monthly income is 4⁸ rupees, he spends 2¹⁰ for monthly electric bill and 215 for other expenses. Rest of the money he saves. Now find the ratio of Rajeev’s savings to his expenses.
    
    Solution:
    Monthly income = 4⁸ = 2¹⁶
    Expenses:
    Electric bill = 2¹⁰
    Other expenese = 2¹⁵
    Total = 2¹⁵ + 2¹⁰
    ∴ Savings = 2¹⁶ - 2¹⁵ - 2¹⁰
    
    
    
    
20. One mole has 6. 022 × 10²³ atoms. Now we take 2.4 moles of A and 1.05 moles of B and mix them together, then find the total number of atoms in the mixture.
    
    Solution:
    1 mole = 6.22 x 10²³ atoms
    Total moles of mixture = 2.4 + 1.05 = 3.45 moles
    ∴ 3.45 moles = 3.34 x 6.022 x 10²³ = 2.07759 x 10²⁴ atoms