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Number - Standard Form (Scientific Notation)

Grade 8Cambridge (IGCSE)

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Standard form (also known as Scientific Notation) is used to write very large or very small numbers concisely.

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A number in standard form must be written in the format a×10na \times 10^n.

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The value of aa must be 1≤a<101 \le a < 10 (a number greater than or equal to 1 but less than 10).

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The exponent nn must be an integer (positive for large numbers, negative for small numbers).

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To convert a large number, move the decimal point to the left until one non-zero digit remains on the left; nn is the number of places moved.

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To convert a small number (less than 1), move the decimal point to the right until it is after the first non-zero digit; nn is the negative of the number of places moved.

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When multiplying numbers in standard form, multiply the coefficients and add the exponents.

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When dividing numbers in standard form, divide the coefficients and subtract the exponents.

📐Formulae

a×10n where 1≤a<10,n∈Za \times 10^n \text{ where } 1 \le a < 10, n \in \mathbb{Z}

(a×10x)×(b×10y)=(a×b)×10x+y(a \times 10^x) \times (b \times 10^y) = (a \times b) \times 10^{x+y}

(a×10x)÷(b×10y)=(a÷b)×10x−y(a \times 10^x) \div (b \times 10^y) = (a \div b) \times 10^{x-y}

💡Examples

Problem 1:

Write 504,000,000 in standard form.

Solution:

5.04×1085.04 \times 10^8

Explanation:

Move the decimal point 8 places to the left to get 5.04. Since we moved left (large number), the exponent is positive 8.

Problem 2:

Write 0.000032 in standard form.

Solution:

3.2×10−53.2 \times 10^{-5}

Explanation:

Move the decimal point 5 places to the right to get 3.2. Since we moved right (small number), the exponent is negative 5.

Problem 3:

Calculate (2×104)×(6×105)(2 \times 10^4) \times (6 \times 10^5), giving your answer in standard form.

Solution:

1.2×10101.2 \times 10^{10}

Explanation:

First, multiply the coefficients: 2×6=122 \times 6 = 12. Then add the powers: 104+5=10910^{4+5} = 10^9. This gives 12×10912 \times 10^9. However, 12 is not between 1 and 10, so convert 12 to 1.2×1011.2 \times 10^1. The final result is 1.2×101×109=1.2×10101.2 \times 10^1 \times 10^9 = 1.2 \times 10^{10}.

Problem 4:

Calculate (4×108)÷(8×103)(4 \times 10^8) \div (8 \times 10^3), giving your answer in standard form.

Solution:

5×1045 \times 10^4

Explanation:

Divide the coefficients: 4÷8=0.54 \div 8 = 0.5. Subtract the exponents: 108−3=10510^{8-3} = 10^5. This gives 0.5×1050.5 \times 10^5. Since 0.5 is less than 1, convert it to 5×10−15 \times 10^{-1}. The final result is 5×10−1×105=5×1045 \times 10^{-1} \times 10^5 = 5 \times 10^4.