CBSE Class 8 Maths: Exponents and Powers Ex 12.2 (Standard Form)

Understanding Standard Form (Scientific Notation)

Standard form is a way of writing numbers as a product of two parts: a number between 1 and 10, and a power of 10. The general format is k × 10ⁿ, where 'k' is a number such that 1 ≤ k < 10, and 'n' is an integer (positive, negative, or zero).

Why is this useful? Imagine the distance from the Earth to the Sun is about 149,600,000,000 meters. Writing all those zeros is tedious and can lead to mistakes. In standard form, we can write this as 1.496 × 10¹¹. It's much cleaner!

Similarly, for very small numbers, like the diameter of a red blood cell which is about 0.000007 meters, we can write it as 7 × 10⁻⁶ meters. The negative exponent tells us it's a very small number. The key is to move the decimal point until there is only one non-zero digit to its left. The number of places you moved the decimal gives you the value of the exponent 'n'.

How to Convert Numbers to Standard Form

1. Step 1: Locate the Decimal Point — Find the decimal point in your number. If it's a whole number like 5985, the decimal point is at the end (5985.).
2. Step 2: Move the Decimal Point — Move the decimal point to the left or right so that there is only ONE non-zero digit to its left. For 5985, you would move it between 5 and 9 to get 5.985. For 0.0032, you would move it between 3 and 2 to get 3.2.
3. Step 3: Count the Jumps — Count how many places you moved the decimal. This number will be your exponent, 'n'.
4. Step 4: Determine the Exponent's Sign — If you moved the decimal to the LEFT (for a large number), the exponent 'n' is POSITIVE. Example: 5985 becomes 5.985. We moved 3 places to the left, so it's 5.985 × 10³. If you moved the decimal to the RIGHT (for a small number), the exponent 'n' is NEGATIVE. Example: 0.0032 becomes 3.2. We moved 3 places to the right, so it's 3.2 × 10⁻³.

Worked Examples: Standard Form to Usual Form

• Example 1: Positive Exponent Express 4.5 × 10⁴ in usual form. Explanation: The exponent is positive 4. This means we need to make the number larger by moving the decimal point 4 places to the right. Steps: Start with 4.5. Move the decimal one place to get 45. We need to move it three more times. We add zeros as placeholders. 4.5 → 45. (1 jump) → 450. (2 jumps) → 4500. (3 jumps) → 45000. (4 jumps) * Answer: 45,000
• Example 2: Negative Exponent Express 3.52 × 10⁻⁵ in usual form. Explanation: The exponent is negative 5. This means we need to make the number smaller by moving the decimal point 5 places to the left. Steps: Start with 3.52. We will need to add leading zeros to move the decimal to the left. 3.52 → .352 (1 jump) → .0352 (2 jumps) → .00352 (3 jumps) → .000352 (4 jumps) → .0000352 (5 jumps) * Answer: 0.0000352

Exam Tips: Watch Out for These Common Errors!

Pay close attention to these points during your exam to avoid losing marks:

1. The 'k' value must be correct: The first number in standard form (k × 10ⁿ) MUST be between 1 and 10 (i.e., 1 ≤ k < 10). Writing 34.5 × 10⁵ is incorrect. You must adjust it to 3.45 × 10⁶.
2. Sign of the Exponent: A simple trick to remember the sign of 'n':

• Very BIG numbers (greater than 1) have POSITIVE exponents.
• Very small numbers (less than 1) have NEGATIVE exponents.

1. Count the Decimal Jumps Carefully: Be patient and double-check the number of places you move the decimal. A common mistake is being off by one, which changes the value of your number completely.

Practice Questions with Solutions

• Q: Express the number 85,200,000,000 in standard form. A: Step 1: The given number is 85,200,000,000. The decimal point is at the end. Step 2: To get a number between 1 and 10, we must move the decimal point between 8 and 5, which gives us 8.52. Step 3: We count the number of places the decimal moved to the left. It moved 10 places. Step 4: Since we moved the decimal to the left for a large number, the exponent is positive. Final answer: 8.52 × 10¹⁰
• Q: Express the number 0.000000061 in standard form. A: Step 1: The given number is 0.000000061. Step 2: To get a number between 1 and 10, we must move the decimal point between 6 and 1, which gives us 6.1. Step 3: We count the number of places the decimal moved to the right. It moved 8 places. Step 4: Since we moved the decimal to the right for a small number, the exponent is negative. Final answer: 6.1 × 10⁻⁸
• Q: Express 3.61492 × 10⁶ in usual form. A: Step 1: The number is 3.61492 × 10⁶. The exponent is +6. Step 2: A positive exponent means we need to make the number larger by moving the decimal point 6 places to the right. Step 3: Moving the decimal in 3.61492 six times to the right: 3.61492 → 36.1492 → 361.492 → 3614.92 → 36149.2 → 361492. → 3614920. Wait, let's recount. We move past the 6, 1, 4, 9, 2 (that's 5 places). We need to move one more place, so we add a zero. Step 4: 3.61492 becomes 3,614,920. Final answer: 3,614,920
• Q: The size of a plant cell is 0.00001275 m. Express it in standard form. A: Step 1: The number is 0.00001275. Step 2: We need to move the decimal point so that it's after the first non-zero digit (1). This gives us 1.275. Step 3: We moved the decimal point from its original position past four zeros and the digit 1. That's a total of 5 places to the right. Step 4: Since we moved the decimal to the right for a small number, the exponent will be negative. Final answer: 1.275 × 10⁻⁵ m

Frequently Asked Questions