Multiply and divide in scientific notation

Product

\(\displaystyle (4\times10^3)(2.5\times10^2)\)

Choose the product in scientific notation

For (a × 10⁴)(b × 10³), compare the raw coefficient product with 10. Try products below, equal to and above 10.

When is the final exponent 8 rather than 7?

Separate, combine, normalise

These examples use positive coefficients. The exponents can be positive, zero or negative

1. 1Multiply coefficients and add indices Deal with the two parts separately \(\displaystyle (3\times10^4)(2\times10^{-6})=6\times10^{-2}\)
2. 2Divide coefficients and subtract indices Subtract the entire denominator exponent, including its sign \(\displaystyle \frac{8\times10^{-3}}{4\times10^{-6}}=2\times10^3\)
3. 3Normalise in the correct direction Dividing the coefficient by 10 increases the exponent by 1; multiplying it by 10 decreases the exponent by 1. \(\displaystyle 0.6\times10^3=6\times10^2\)
4. 4Round only at the end Keep the exact value until rounding is requested. This example is rounded to two significant figures \(\displaystyle 1.764\times10^1\approx1.8\times10^1\)

Given

\(\displaystyle (8\times10^6)(7.5\times10^3)\)

Find the exponent of the final normalised answer

Given

\(\displaystyle \frac{1.5\times10^8}{2.5\times10^5}\)

Find the coefficient of the final normalised answer

A calculation gives 9.96 × 10⁵. Choose the result to two significant figures in scientific notation