Multiplying and Dividing Numbers in Scientific Notation

Scientific notation writes a number as a coefficient times a power of ten, usually with a coefficient from 1 up to—but not including—10. It is useful for very large and very small values. Multiplication and division become manageable because powers of ten follow exponent rules.

Multiply: multiply coefficients, add exponents

For (a × 10^m)(b × 10^n), calculate (a × b) × 10^(m+n). Example: (3 × 10^4)(2 × 10^−3) = 6 × 10^1 = 60. The coefficient multiplication and exponent addition are separate steps. Keep the exponent signs visible so a negative exponent does not become an accidental subtraction error.

Divide: divide coefficients, subtract exponents

For (a × 10^m) ÷ (b × 10^n), calculate (a ÷ b) × 10^(m−n). Example: (8 × 10^6) ÷ (2 × 10^2) = 4 × 10^4. When subtracting a negative exponent, add its positive value: 10^3 ÷ 10^−2 = 10^(3−(−2)) = 10^5.

Normalize the coefficient

If multiplication gives a coefficient of 18, rewrite 18 × 10^5 as 1.8 × 10^6 by moving the decimal one place left and increasing the exponent by one. If division gives 0.4 × 10^3, rewrite it as 4 × 10^2 by moving the decimal one place right and reducing the exponent by one. The value stays the same.

Check the magnitude

• Estimate whether multiplying should make the value larger or smaller.
• Check that coefficient multiplication or division is correct.
• Write the exponent operation explicitly before simplifying.
• Normalize the coefficient and verify the decimal shift changes the exponent in the opposite direction.

Show the exponent operation separately

Write the coefficient calculation and the power-of-ten calculation on separate parts of a line. For division, (6 × 10⁻³)/(2 × 10⁵) = (6/2) × 10⁻³⁻⁵ = 3 × 10⁻⁸. The divisor's exponent is subtracted as a whole number, including its sign. A common error is to compute −3 − 5 as 2 or to add the exponents in a division problem.

Normalize without changing the value

After multiplying or dividing coefficients, make sure the coefficient has absolute value at least 1 and less than 10. If a product is 42 × 10⁶, rewrite it as 4.2 × 10⁷: moving the coefficient decimal one place left requires raising the exponent by one to preserve the same value. If a quotient is 0.25 × 10⁻⁴, rewrite it as 2.5 × 10⁻⁵. Moving the decimal right lowers the exponent.

Addition and subtraction need a different step

The multiplication and division shortcuts do not apply to addition. To add 3 × 10⁴ and 5 × 10³, rewrite the second term as 0.5 × 10⁴ and then add the coefficients: 3.5 × 10⁴. Alternatively, write both numbers in standard form, add, and convert back. Only coefficients with the same power of ten can be combined directly. Keep track of place value so a decimal shift does not alter the sign or size accidentally.

Use magnitude as a check

A quick estimate can reveal an exponent mistake. Multiplying a number near 10⁶ by one near 10² should produce a result near 10⁸. Dividing a number near 10⁶ by one near 10² should produce a result near 10⁴. If your normalized answer is near 10⁻⁴ instead, revisit the exponent arithmetic. Also check whether the coefficient product or quotient is sensible before normalizing.

Keep signs distinct

The sign of the coefficient controls whether the value is positive or negative; the exponent controls its scale. For example, (−2 × 10³)(4 × 10⁻²) = −8 × 10¹ = −80. The negative sign does not change the exponent rule. In a division problem, remember that a nonzero denominator is required, and divide the coefficients before normalizing.

Key takeaway

Multiply coefficients and add exponents; divide coefficients and subtract exponents. Normalize at the end and use an estimate to catch sign and decimal errors.

Multiply coefficients and combine powers

Scientific notation writes a number as a coefficient times a power of ten, with a coefficient at least 1 and less than 10 in absolute value: 4.2 × 10⁵. To multiply, multiply the coefficients and add exponents: (3 × 10⁴)(2 × 10³) = 6 × 10⁷. Then check that the coefficient remains in the required range.

For (8 × 10⁶)(5 × 10²), multiply to get 40 × 10⁸. Normalize 40 as 4 × 10¹, so the result is 4 × 10⁹. A frequent error is to stop at 40 × 10⁸ even though 40 is not between 1 and 10. Moving the decimal one place left requires increasing the exponent by one to keep the value unchanged.

Divide coefficients and subtract exponents

To divide powers with the same base, subtract exponents: (a × 10ᵐ)/(b × 10ⁿ) = (a/b) × 10ᵐ⁻ⁿ. For (9 × 10⁷)/(3 × 10²), divide coefficients to get 3 and subtract exponents to get 10⁵, so the result is 3 × 10⁵. Keep track of which exponent is in the numerator and which is in the denominator.

A negative exponent represents a positive fraction less than 1: 10⁻³ = 1/1000. For (4 × 10³)/(8 × 10⁵), coefficient division gives 0.5 and exponent subtraction gives 10⁻²; the product is 0.005, which normalizes to 5 × 10⁻³. Both decimal and scientific forms provide a useful reasonableness check.

Normalize and compare magnitudes

After multiplying or dividing, rewrite the coefficient so its decimal point is one place after the first nonzero digit. If normalization moves the decimal left, increase the exponent; if it moves right, decrease the exponent. For example, 0.72 × 10⁴ becomes 7.2 × 10³. The value remains 7,200.

For addition and subtraction, exponents must first match; align powers of ten before combining coefficients. This differs from multiplication and division, where exponents are added or subtracted. If a question mixes operations, follow order of operations and normalize only as needed after each stage.

• Multiply coefficients and add exponents when multiplying powers of ten.
• Divide coefficients and subtract exponents when dividing.
• Normalize the coefficient to the range from 1 up to but not including 10.
• Use a negative exponent to represent a small positive value.
• For addition or subtraction, align exponents first.

Apply it to the evidence or sentence

In scientific notation, multiply the decimal coefficients and add the powers of ten; divide the coefficients and subtract the exponents. For (3 × 10⁴)(2 × 10³), the result is 6 × 10⁷. For (8 × 10⁶) ÷ (4 × 10²), it is 2 × 10⁴. Normalize the coefficient so it is at least 1 and less than 10: 24 × 10⁵ becomes 2.4 × 10⁶. Track the exponent sign carefully, especially when dividing by a small number represented with a negative exponent. Estimate the magnitude first to catch an answer that is off by a power of ten.