CHAPTER 10 - EXPONENTS AND POWERS

Exercise 10.2

1. Express the following numbers in standard form.
(i) \(0.0000000000085\)

To write in standard form, move the decimal point to the right until there is one non-zero digit to its left. The number of places moved becomes the negative exponent of 10.

The decimal point is moved 12 places to the right.

\[ 0.0000000000085 = 8.5 \times 10^{-12} \]

Final answer: \(8.5 \times 10^{-12}\)
(ii) \(0.00000000000942\)

Move the decimal point 12 places to the right so that there is a single non-zero digit to the left of the decimal.

\[ 0.00000000000942 = 9.42 \times 10^{-12} \]

Final answer: \(9.42 \times 10^{-12}\)
(iii) \(6020000000000000\)

To write in standard form, move the decimal point to the left until there is one non-zero digit to its left. The number of places moved becomes the positive exponent of 10.

The decimal point is moved 15 places to the left.

\[ 6020000000000000 = 6.02 \times 10^{15} \]

Final answer: \(6.02 \times 10^{15}\)
(iv) \(0.00000000837\)

Move the decimal point 9 places to the right.

\[ 0.00000000837 = 8.37 \times 10^{-9} \]

Final answer: \(8.37 \times 10^{-9}\)
(v) \(31860000000\)

Move the decimal point 10 places to the left.

\[ 31860000000 = 3.186 \times 10^{10} \]

Final answer: \(3.186 \times 10^{10}\)

2. Express the following numbers in usual form.
(i) \(3.02 \times 10^{-6}\)

A negative exponent means the number is less than 1. Move the decimal point to the left by the number of places indicated by the exponent.

\[ 3.02 \times 10^{-6} = 0.00000302 \]

Final answer: \(0.00000302\)
(ii) \(4.5 \times 10^{4}\)

A positive exponent means the number is greater than 10. Move the decimal point to the right by the number of places indicated by the exponent.

\[ 4.5 \times 10^{4} = 45000 \]

Final answer: \(45000\)
(iii) \(3 \times 10^{-8}\)

Move the decimal point 8 places to the left.

\[ 3 \times 10^{-8} = 0.00000003 \]

Final answer: \(0.00000003\)
(iv) \(1.0001 \times 10^{9}\)

Move the decimal point 9 places to the right.

\[ 1.0001 \times 10^{9} = 1000100000 \]

Final answer: \(1000100000\)
(v) \(5.8 \times 10^{12}\)

Move the decimal point 12 places to the right.

\[ 5.8 \times 10^{12} = 5800000000000 \]

Final answer: \(5800000000000\)
(vi) \(3.61492 \times 10^{6}\)

Move the decimal point 6 places to the right.

\[ 3.61492 \times 10^{6} = 3614920 \]

Final answer: \(3614920\)

3. Express the number appearing in the following statements in standard form.
(i) 1 micron is equal to \(\frac{1}{1000000}\) m.

\[ \frac{1}{1000000} \text{ m} = \frac{1}{10^6} \text{ m} = 1 \times 10^{-6} \text{ m} \]

Final answer: \(1 \times 10^{-6}\) m
(ii) Charge of an electron is \(0.00000000000000000016\) coulomb.

Move the decimal point 19 places to the right.

\[ 0.00000000000000000016 \text{ C} = 1.6 \times 10^{-19} \text{ C} \]

Final answer: \(1.6 \times 10^{-19}\) C
(iii) Size of a bacteria is \(0.0000005\) m.

Move the decimal point 7 places to the right.

\[ 0.0000005 \text{ m} = 5 \times 10^{-7} \text{ m} \]

Final answer: \(5 \times 10^{-7}\) m
(iv) Size of a plant cell is \(0.00001275\) m.

Move the decimal point 5 places to the right.

\[ 0.00001275 \text{ m} = 1.275 \times 10^{-5} \text{ m} \]

Final answer: \(1.275 \times 10^{-5}\) m
(v) Thickness of a thick paper is \(0.07\) mm.

Move the decimal point 2 places to the right.

\[ 0.07 \text{ mm} = 7 \times 10^{-2} \text{ mm} \]

Final answer: \(7 \times 10^{-2}\) mm

4. In a stack there are 5 books each of thickness \(20\) mm and 5 paper sheets each of thickness \(0.016\) mm. What is the total thickness of the stack?

First, find the total thickness of the books.

\[ \text{Thickness of 5 books} = 5 \times 20 \text{ mm} = 100 \text{ mm} \]

Next, find the total thickness of the paper sheets.

\[ \text{Thickness of 5 sheets} = 5 \times 0.016 \text{ mm} = 0.08 \text{ mm} \]

Add the two thicknesses to find the total thickness.

\[ \text{Total thickness} = 100 \text{ mm} + 0.08 \text{ mm} = 100.08 \text{ mm} \]

Express the answer in standard form as requested in the overall exercise.

\[ 100.08 \text{ mm} = 1.0008 \times 10^2 \text{ mm} \]

\(1.0008 \times 10^{2}\) mm

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