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Number and Algebra - Scientific notation

Grade 11IB_AI

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

🔑Concepts

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Scientific notation, also known as standard form, is a way of writing very large or very small numbers efficiently in the form a×10ka \times 10^k.

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The value of aa (the mantissa) must be greater than or equal to 1 and strictly less than 10 (1≤a<101 \le a < 10).

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The exponent kk must be an integer (k∈Zk \in \mathbb{Z}). A positive kk represents a number larger than 10, while a negative kk represents a number between 0 and 1.

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When converting from standard decimal form to scientific notation, if you move the decimal point to the left, kk is positive. If you move it to the right, kk is negative.

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IB AI exams often require answers to be rounded to 3 significant figures unless specified otherwise, even when using scientific notation.

📐Formulae

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

(a×10m)×(b×10n)=(a×b)×10m+n(a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{m+n}

a×10mb×10n=(ab)×10m−n\frac{a \times 10^m}{b \times 10^n} = \left(\frac{a}{b}\right) \times 10^{m-n}

💡Examples

Problem 1:

The distance from the Earth to the Sun is approximately 149,600,000149,600,000 km. Express this distance in scientific notation to 3 significant figures.

Solution:

1.50×1081.50 \times 10^8 km

Explanation:

First, identify the first three significant figures: 1,4,91, 4, 9. The fourth digit is 66, so we round up the 99 to 1010, making it 1.501.50. We move the decimal point 8 places to the left to get a number between 1 and 10, so the exponent is 88.

Problem 2:

Calculate the value of (4.2×10−3)÷(2.0×10−7)(4.2 \times 10^{-3}) \div (2.0 \times 10^{-7}). Give your answer in scientific notation.

Solution:

2.1×1042.1 \times 10^4

Explanation:

Divide the coefficients: 4.2÷2.0=2.14.2 \div 2.0 = 2.1. Subtract the exponents for division: −3−(−7)=−3+7=4-3 - (-7) = -3 + 7 = 4. Combine the results to get 2.1×1042.1 \times 10^4.

Problem 3:

A microscopic organism has a length of 0.00000008020.0000000802 meters. Write this length in scientific notation.

Solution:

8.02×10−88.02 \times 10^{-8} m

Explanation:

To obtain a coefficient aa such that 1≤a<101 \le a < 10, we move the decimal point 8 places to the right. Because we moved the decimal to the right (signifying a very small value), the exponent is negative, resulting in 8.02×10−88.02 \times 10^{-8}.