GSEB Class 7 Maths Solutions Chapter 13 Exponents and Powers Exercise 13.3

Question 1. Write the numbers in the expanded forms:
(i) 279404
(ii) 3006194
(iii) 2806196
(iv) 120719
(v) 20068
Answer:
(i) For the number 279404, we can expand it as follows:
\( 279404 = 2 \times 100000 + 7 \times 10000 + 9 \times 1000 + 4 \times 100 + 0 \times 10 + 4 \times 1 \)
\( = 2 \times 10^5 + 7 \times 10^4 + 9 \times 10^3 + 4 \times 10^2 + 0 \times 10^1 + 4 \times 10^0 \)
\( = 2 \times 10^5 + 7 \times 10^4 + 9 \times 10^3 + 4 \times 10^2 + 4 \times 10^0 \)
(ii) For the number 3006194, we can expand it as follows:
\( 3006194 = 3 \times 1000000 + 0 \times 100000 + 0 \times 10000 + 6 \times 1000 + 1 \times 100 + 9 \times 10 + 4 \times 1 \)
\( = 3 \times 10^6 + 0 \times 10^5 + 0 \times 10^4 + 6 \times 10^3 + 1 \times 10^2 + 9 \times 10^1 + 4 \times 10^0 \)
\( = 3 \times 10^6 + 6 \times 10^3 + 1 \times 10^2 + 9 \times 10^1 + 4 \times 10^0 \)
(iii) For the number 2806196, we can expand it as follows:
\( 2806196 = 2 \times 1000000 + 8 \times 100000 + 0 \times 10000 + 6 \times 1000 + 1 \times 100 + 9 \times 10 + 6 \times 10^0 \)
\( = 2 \times 10^6 + 8 \times 10^5 + 0 \times 10^4 + 6 \times 10^3 + 1 \times 10^2 + 9 \times 10^1 + 6 \times 10^0 \)
\( = 2 \times 10^6 + 8 \times 10^5 + 6 \times 10^3 + 1 \times 10^2 + 9 \times 10^1 + 6 \times 10^0 \)
(iv) For the number 120719, we can expand it as follows:
\( 120719 = 1 \times 100000 + 2 \times 10000 + 0 \times 1000 + 7 \times 100 + 1 \times 10 + 9 \times 1 \)
\( = 1 \times 10^5 + 2 \times 10^4 + 0 \times 10^3 + 7 \times 10^2 + 1 \times 10^1 + 9 \times 10^0 \)
\( = 1 \times 10^5 + 2 \times 10^4 + 7 \times 10^2 + 1 \times 10^1 + 9 \times 10^0 \)
(v) For the number 20068, we can expand it as follows:
\( 20068 = 2 \times 10000 + 0 \times 1000 + 0 \times 100 + 6 \times 10 + 8 \times 1 \)
\( = 2 \times 10^4 + 0 \times 10^3 + 0 \times 10^2 + 6 \times 10^1 + 8 \times 10^0 \)
\( = 2 \times 10^4 + 6 \times 10^1 + 8 \times 10^0 \)
In simple words: To write a number in expanded form, break it down by place value. Each digit is multiplied by a power of 10, where the power matches its position.

Exam Tip: Remember that any number raised to the power of 0 is 1. Be careful with zeros in the number; they still need their place value term (e.g., \(0 \times 10^2\)).

Question 2. Find the number from each of the following expanded forms:
(a) \( 8 \times 10^4 + 6 \times 10^3 + 0 \times 10^2 + 4 \times 10^1 + 5 \times 10^0 \)
(b) \( 4 \times 10^5 + 5 \times 10^3 + 3 \times 10^2 + 2 \times 10^0 \)
(c) \( 3 \times 10^4 + 7 \times 10^2 + 5 \times 10^0 \)
(d) \( 9 \times 10^5 + 2 \times 10^2 + 3 \times 10^1 \)
Answer:
(a) We can find the number from the expanded form:
\( 8 \times 10^4 + 6 \times 10^3 + 0 \times 10^2 + 4 \times 10^1 + 5 \times 10^0 \)
\( = 8 \times 10000 + 6 \times 1000 + 0 \times 100 + 4 \times 10 + 5 \times 1 \)
\( = 80000 + 6000 + 0 + 40 + 5 \)
\( = 86045 \)
(b) We can find the number from the expanded form:
\( 4 \times 10^5 + 5 \times 10^3 + 3 \times 10^4 + 2 \times 10^0 \)
\( = 4 \times 100000 + 5 \times 1000 + 3 \times 100 + 2 \times 1 \)
\( = 400000 + 5000 + 300 + 2 \)
\( = 405302 \)
(c) We can find the number from the expanded form:
\( 3 \times 10^4 + 7 \times 10^2 + 5 \times 10^0 \)
\( = 3 \times 10000 + 7 \times 100 + 5 \times 1 \)
\( = 30000 + 700 + 5 \)
\( = 30705 \)
(d) We can find the number from the expanded form:
\( 9 \times 10^5 + 2 \times 10^2 + 3 \times 10^1 \)
\( = 9 \times 100000 + 2 \times 100 + 3 \times 10 \)
\( = 900000 + 200 + 30 \)
\( = 900230 \)
In simple words: To get the number from its expanded form, multiply each part and then add all the results together. This will give you the original number.

Exam Tip: Pay close attention to the powers of ten. If a power is missing, it means its coefficient is zero, which is important for correctly placing the digits.

Question 3. Express the following numbers in standard form:
(i) 5,00,00,000
(ii) 70,00,000
(iii) 3,18,65,00,000
(iv) 3,90,878
(v) 39087.8
(vi) 3908.78
Answer:
(i) For the number 5,00,00,000:
We have: \( 50000000 = 5 \times 10000000 = 5 \times 10^7 \)
Thus, the standard form of 50000000 is \( 5 \times 10^7 \).
(ii) For the number 70,00,000:
We have: \( 7000000 = 7 \times 1000000 = 7 \times 10^6 \)
Thus, the standard form of 7000000 is \( 7 \times 10^6 \).
(iii) For the number 3,18,65,00,000:
We have: \( 3186500000 = 3.1865 \times 10^9 \)
Thus, the standard form of 3186500000 is \( 3.1865 \times 10^9 \).
(iv) For the number 3,90,878:
We have: \( 390878 = 3.90878 \times 10^5 \)
Thus, the standard form of 390878 is \( 3.90878 \times 10^5 \).
(v) For the number 39087.8:
We have: \( 39087.8 = 3.90878 \times 10^4 \)
Thus, the standard form of 39087.8 is \( 3.90878 \times 10^4 \).
(vi) For the number 3908.78:
We have: \( 3908.78 = 3.90878 \times 10^3 \)
Thus, the standard form of 3908.78 is \( 3.90878 \times 10^3 \).
In simple words: To write a number in standard form, you move the decimal point until there is only one non-zero digit before it. The count of places you moved the decimal becomes the exponent of 10.

Exam Tip: Remember to express the number as a decimal between 1 and 10 (inclusive of 1, exclusive of 10) multiplied by a power of 10. The exponent is positive for large numbers and negative for small numbers.

Question 4. Express the number appearing in the following statements in standard form.
(a) The distance between Earth and Moon is 384,000,000 m.
(b) Speed of light in vacuum is 300,000,000 m/s.
(c) Diameter of the Earth is 1,27,56,000 m.
(d) Diameter of the Sun is 1,400,000,000 m.
(e) In a galaxy there are on an average 100,000,000,000 stars.
(f) The universe is estimated to be about 12,000,000,000 years old.
(g) The distance of the Sun from the centre of the Milky Way Galaxy is estimated to be 300,000,000,000,000,000,000 m.
(h) 60,230,000,000,000,000,000,000 molecules are contained in a drop of water weighing 1.8 gm.
(i) The earth has 1,353,000,000 cubic km of sea water.
(j) The population of India was about 1,027,000,000 in March, 2001.
Answer:
(a) The distance between Earth and Moon is 384,000,000 m.
In standard form: \( 384,000,000 = 3.84 \times 10^8 \) m.
(b) The speed of light in vacuum is 300,000,000 m/s.
In standard form: \( 300,000,000 = 3 \times 10^8 \) m/s.
(c) The diameter of the Earth is 1,27,56,000 m.
In standard form: \( 1,27,56,000 = 1.2756 \times 10^7 \) m.
(d) The diameter of the Sun is 1,400,000,000 m.
In standard form: \( 1,400,000,000 = 1.4 \times 10^9 \) m.
(e) In a galaxy, there are on average 100,000,000,000 stars.
In standard form: \( 100,000,000,000 = 1 \times 10^{11} \) stars.
(f) The universe is estimated to be about 12,000,000,000 years old.
In standard form: \( 12,000,000,000 = 1.2 \times 10^{10} \) years old.
(g) The distance of the Sun from the centre of the Milky Way Galaxy is 300,000,000,000,000,000,000 m.
In standard form: \( 300,000,000,000,000,000,000 = 3 \times 10^{20} \) m.
(h) 60,230,000,000,000,000,000,000 molecules are contained in a drop of water weighing 1.8 gm.
In standard form: \( 60,230,000,000,000,000,000,000 = 6.023 \times 10^{22} \) molecules.
(i) The Earth has 1,353,000,000 cubic km of sea water.
In standard form: \( 1,353,000,000 = 1.353 \times 10^9 \) cubic km.
(j) The population of India was about 1,027,000,000 in March, 2001.
In standard form: \( 1,027,000,000 = 1.027 \times 10^9 \).
In simple words: To change a large number into standard form, you move the decimal point until there's just one digit left before it. Then, you multiply that number by 10 raised to the power of how many places you moved the decimal.

Exam Tip: Accurately count the number of places the decimal point is moved to determine the correct exponent. Ensure the first part of the standard form (the coefficient) is a number between 1 and 10.