RBSE Solutions Class 8 Maths Chapter 3 Powers and Exponents Exercise 3.3

Question 1. Change In the standard form
(i) 128000000
(ii) 1680000000
(iii) 0.0005
(iv) 0.00000017
(v) 0.000000000397
(vi) 0.00000004358
Answer:
(i) \( 128000000 = 1.28 \times 100000000 = 1.28 \times 10^8 \)
(ii) \( 1680000000 = 1.68 \times 1000000000 = 1.68 \times 10^9 \)
(iii) \( 0.0005 = 5 \times 10^{-4} \)
(iv) \( 0.00000017 = 1.7 \times \frac{1}{10^7} = 1.7 \times 10^{-7} \)
(v) \( 0.000000000397 = \frac{397}{10^{12}} = \frac{397}{10^2 \times 10^{10}} = \frac{3.97}{10^{10}} = 3.97 \times 10^{-10} \)
(vi) \( 0.00000004358 = \frac{4358}{10^{11}} = \frac{4358}{10^3 \times 10^8} = \frac{4.358}{10^8} = 4.358 \times 10^{-8} \)
In simple words: To write a number in standard form, you move the decimal point until there's only one non-zero digit before it. Then, you multiply by a power of 10. The exponent shows how many places the decimal point was moved and in which direction.

🎯 Exam Tip: Remember that for large numbers (like 128,000,000), the exponent of 10 will be positive, and for small numbers (like 0.0005), the exponent will be negative.

Question 2. Express the following numbers in normal form
(i) \( 4 \times 10^9 \)
(ii) \( 245 \times 10^7 \)
(iii) \( 5.61729 \times 10^7 \)
(iv) \( 8.5 \times 10^{-6} \)
(v) \( 3.02 \times 10^{-6} \)
(vi) \( 7 \times 10^{-4} \)
Answer:
(i) \( 4 \times 10^9 = 4 \times 1000000000 = 4000000000 \)
(ii) \( 245 \times 10^7 = 245 \times 10000000 = 2450000000 \)
(iii) \( 5.61729 \times 10^7 = 56172900 \)
(iv) \( 8.5 \times 10^{-6} = 0.0000085 \)
(v) \( 3.02 \times 10^{-6} = 0.00000302 \)
(vi) \( 7 \times 10^{-4} = 0.0007 \)
In simple words: To change a number from standard form to normal form, you move the decimal point. If the power of 10 is positive, move the decimal to the right. If the power of 10 is negative, move the decimal to the left.

🎯 Exam Tip: Be careful when counting the number of places to move the decimal point; a common error is to miscount the zeros.

Question 3. Change the numbers In standard form of the following statements
(i) Diameter of the thickness of human hair is 0.0002 cm approx.
(ii) Charge on an electron is 0.000,000,000,000,000,00016 coulomb.
(iii) 1 Micron = \( \frac {1}{1000000 } \) metre.
(iv) Thickness of a paper = 0.0016 cm.
Answer:
(i) \( 0.0002 \text{ cm} = 2 \times 10^{-4} \text{ cm} \)
(ii) \( 0.00000000000000000016 \text{ coulomb} = 1.6 \times 10^{-19} \text{ coulomb} \)
(iii) \( 1 \text{ Micron} = \frac{1}{1000000} \text{ metre} = 1 \times 10^{-6} \text{ metre} \)
(iv) \( 0.0016 \text{ cm} = \frac{16}{10000} \text{ cm} = \frac{16}{10^4} \text{ cm} = 16 \times 10^{-4} \text{ cm} = 1.6 \times 10^{-3} \text{ cm} \)
In simple words: Standard form helps us write very big or very small numbers in a shorter, easier way. It uses powers of 10 to show how many decimal places are involved. This makes scientific values much simpler to read.

🎯 Exam Tip: Always remember to include the correct units (like cm, coulomb, or metre) in your final answer when dealing with word problems.