CBSE Class 7 Maths: Exponents and Powers - Exercise 13.3 Explained

What is Standard Form (Scientific Notation)?

Imagine trying to write the distance from the Earth to the Sun, which is approximately 150,000,000,000 metres. That's a lot of zeros to count and keep track of! 'Standard form' (also known as scientific notation) is a special way of writing very large or very small numbers using powers of 10. It makes these numbers much easier to read, write, and compare. The general form of a number in standard form is \(k \times 10^n\), where 'k' is a number between 1 and 10 (including 1, but not 10), and 'n' is an integer (a whole number, which can be positive or negative). For Class 7, we will mostly focus on large numbers, so 'n' will typically be a positive integer. This method simplifies incredibly complex figures into a neat, two-part expression, significantly reducing the chances of errors and improving clarity. It's like having a superpower for numbers!

How to Write a Number in Standard Form

1. Step 1: Locate the Decimal Point — For whole numbers, the decimal point is always at the very end, even if it's not written. For example, in 5,000,000, the decimal point is after the last zero (5,000,000.).
2. Step 2: Move the Decimal Point — Shift the decimal point to the left until there is only one non-zero digit remaining to its left. This digit should be between 1 and 9 (inclusive). For example, 5,000,000. becomes 5.000000.
3. Step 3: Count the Shifts — Count how many places you moved the decimal point. This count will be your exponent 'n'. If you moved the decimal to the left (for large numbers), the exponent 'n' will be positive. If you were working with a very small number and moved it to the right, 'n' would be negative.
4. Step 4: Write in Standard Form — Combine the new number (from Step 2, which is your 'k' value) with the power of 10 (from Step 3). So, it will be \(k \times 10^n\). For 5,000,000, we moved the decimal 6 places to the left, so it becomes \(5 \times 10^6\).

Comparing Numbers Written in Standard Form

Once you can write numbers in standard form, comparing them becomes much simpler than counting endless zeros! To compare two numbers written in standard form, say \(A = k_1 \times 10^{n_1}\) and \(B = k_2 \times 10^{n_2}\), you follow a simple two-step process. First, look at the exponents of 10 (\(n_1\) and \(n_2\)). The number with the larger exponent is always the larger number. For instance, \(5 \times 10^7\) is greater than \(8 \times 10^6\) because \(7 > 6\), even though \(5 < 8\). The power of 10 determines the 'magnitude' or 'size' of the number much more significantly than the 'k' part. Only if the exponents are the same (i.e., \(n_1 = n_2\)) do you then compare the 'k' parts (\(k_1\) and \(k_2\)). The number with the larger 'k' value will be the larger number in that case. For example, \(3.2 \times 10^5\) is greater than \(1.9 \times 10^5\) because \(3.2 > 1.9\) and their exponents are identical. This systematic approach ensures accurate comparison of even vastly different quantities.

YoLearn AI Tutor's Exam Tip for Exponents and Powers Ex 13.3

When converting large numbers to standard form, remember that the 'k' value MUST be between 1 (inclusive) and 10 (exclusive). This means \(1.23 \times 10^5\) is correct, but \(12.3 \times 10^4\) is not in standard form, even though it represents the same number. Always ensure your decimal point is placed after the first non-zero digit. For example, for 56,000,000, it should be \(5.6 \times 10^7\), not \(0.56 \times 10^8\) or \(56 \times 10^6\). Also, for numbers greater than 1, your exponent 'n' will always be a positive integer.

Practice Questions with Solutions

• Q: Express the number 7,890,000,000 in standard form. A: Step 1: Identify the implied decimal point at the end of the number: 7,890,000,000. Step 2: Move the decimal point to the left until there is only one non-zero digit before it: 7.890000000. Step 3: Count the number of places the decimal point was moved. It was moved 9 places to the left. Step 4: Write the number in standard form: \(7.89 \times 10^9\). Final answer: \(7.89 \times 10^9\)
• Q: Write the number 305,000,000,000 in standard form. A: Step 1: The decimal point is at the end: 305,000,000,000. Step 2: Move the decimal point left to get one non-zero digit before it: 3.05000000000. Step 3: Count the shifts. The decimal moved 11 places to the left. Step 4: Express in standard form: \(3.05 \times 10^{11}\). Final answer: \(3.05 \times 10^{11}\)
• Q: Compare the numbers \(4.2 \times 10^8\) and \(9.1 \times 10^7\). Which one is larger? A: Step 1: Compare the exponents of 10. For \(4.2 \times 10^8\), the exponent is 8. For \(9.1 \times 10^7\), the exponent is 7. Step 2: Since \(8 > 7\), the number with the larger exponent is the larger number. Final answer: \(4.2 \times 10^8\) is larger than \(9.1 \times 10^7\).
• Q: The population of a city is approximately 25,000,000. Write this number in standard form. A: Step 1: The number is 25,000,000. The decimal point is implied at the end. Step 2: Move the decimal point left until there is one non-zero digit before it: 2.5000000. Step 3: Count the number of places the decimal was moved. It was moved 7 places to the left. Step 4: Write in standard form: \(2.5 \times 10^7\). Final answer: \(2.5 \times 10^7\)

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