Exponents & Roots • Topic 5 of 5

Scientific Notation

Scientific notation writes a number as a × 10ⁿ with the coefficient a between 1 and 10. Big numbers get a positive power (300,000 = 3 × 10⁵); small numbers get a negative power (0.00042 = 4.2 × 10⁻⁴). To multiply, multiply the coefficients and add the exponents; to divide, divide the coefficients and subtract the exponents — then, if the coefficient drifts outside 1–10, renormalise by shifting the decimal and adjusting the power. This is how the GRE handles very large or very small figures (populations, wavelengths, probabilities) without an unwieldy string of zeros, and it makes order-of-magnitude comparisons trivial: 6 × 10⁷ is a thousand times 6 × 10⁴ because the exponents differ by 3. On Quantitative Comparison, comparing two numbers in scientific notation usually comes down to comparing the powers of 10 first, and only checking the coefficients when the powers tie.

✅ Solved examples

1. Write 0.00056 in scientific notation.
Move the decimal 4 places right to get 5.6: 0.00056 = 5.6 × 10⁻⁴.
2. Compute (3 × 10⁵)(2 × 10³).
Multiply coefficients, add exponents: (3·2) × 10⁵⁺³ = 6 × 10⁸.
3. Compute (8 × 10⁷) / (2 × 10³).
Divide coefficients, subtract exponents: 4 × 10⁴.
4. Compute (5 × 10⁶)(4 × 10⁻²) and give the answer in proper scientific notation.
5·4 = 20 and 10⁶⁺⁽⁻²⁾ = 10⁴, giving 20 × 10⁴. Renormalise: 2.0 × 10⁵.

✏️ Practice — try these, take hints as needed

1. Write 74,000 in scientific notation.
Coefficient between 1 and 10.
Move the decimal 4 places.
7.4 × 10^?
7.4 × 10⁴
2. Write 0.0000091 in scientific notation.
Small number ⇒ negative power.
Move decimal 6 places right.
9.1 × 10^?
9.1 × 10⁻⁶
3. Compute (4 × 10⁶)(2 × 10⁵).
Multiply coefficients.
Add exponents.
8 × 10^(6+5).
8 × 10¹¹
4. Compute (9 × 10⁸) / (3 × 10²).
Divide coefficients.
Subtract exponents.
3 × 10^(8−2).
3 × 10⁶
5. Compute (6 × 10⁴)(5 × 10³) in proper scientific notation.
6·5 = 30.
10⁴⁺³ = 10⁷.
Renormalise 30 × 10⁷.
3 × 10⁸

📝 Topic test — 8 questions

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