Indices and scientific notation
Express very large or small numbers and calculate with powers of ten.
Before you begin
- Multiplication
- Decimal place value
Apply index laws with a shared base
Repeated multiplication explains why aᵐ × aⁿ = aᵐ⁺ⁿ: you combine m factors and n factors. For a nonzero base, division subtracts exponents and a negative exponent means a reciprocal. These laws concern multiplication and division, not addition. For example, 10² + 10³ is 1,100, not 10⁵.
Normalise scientific notation
Write a positive number as a × 10ⁿ with 1 ≤ a < 10 and integer n. Moving a decimal point changes the coefficient, so compensate using the power of ten. In a product, multiply coefficients and add exponents, then adjust the coefficient back into the allowed interval. Small positive numbers have negative exponents.
Worked example
Calculate (3 × 10⁴)(4 × 10⁻²) in scientific notation.
Show the worked solution
- Multiply the coefficients: 3 × 4 = 12.
- Add the exponents: 4 + (−2) = 2, giving 12 × 10².
- Rewrite the coefficient as 1.2 and increase the exponent by one.
1.2 × 10³.
Try it yourself
Which expression equals 10⁻³?
Common mistakes
- A coefficient of 12 is not in normalised scientific notation.
What can you explain now?
Write 0.00056 in scientific notation.
Compare with the explanation
5.6 × 10⁻⁴.
The coefficient 5.6 is between 1 and 10. Multiplying it by one ten-thousandth gives the original value.
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Common Core mathematics standards ↗