Standard Form (Multiply & Divide)

Statement

Standard form (or scientific notation) expresses numbers as \(a \times 10^n\), where \(1 \leq a < 10\) and \(n\) is an integer. To multiply or divide numbers in standard form, we work separately with the front numbers (coefficients) and the powers of ten.

\[ (a \times 10^m)(b \times 10^n) = (ab) \times 10^{m+n}, \quad \frac{a \times 10^m}{b \times 10^n} = \left(\frac{a}{b}\right) \times 10^{m-n} \]

Why it’s true

• Multiplying powers of ten: \(10^m \times 10^n = 10^{m+n}\).
• Dividing powers of ten: \(10^m \div 10^n = 10^{m-n}\).
• The coefficients \(a\) and \(b\) are multiplied or divided in the usual way.
• If the result is not between 1 and 10, adjust the coefficient and power of ten to return to standard form.

Recipe (how to use it)

1. Multiply or divide the front numbers \(a\) and \(b\).
2. Add powers when multiplying, subtract powers when dividing.
3. Check the coefficient: if it is less than 1 or 10 or more, rewrite it into correct standard form by adjusting the power of ten.

Spotting it

Look for questions with very large or very small numbers, often written in \( \times 10^n \) form. These are common in physics and astronomy problems.

Common pairings

• Using standard form with units like speed of light, population, or particle size.
• Working with calculators that give results in scientific notation.

Mini examples

1. Given: \((3 \times 10^4)(2 \times 10^3)\). Answer: \(6 \times 10^7\).
2. Given: \((8 \times 10^6) \div (2 \times 10^2)\). Answer: \(4 \times 10^4\).

Pitfalls

• Forgetting to adjust the coefficient into the range 1 ≤ a < 10.
• Mixing up addition and multiplication of indices.
• Dropping negative signs when powers are subtracted.
• Leaving answers not in standard form.

Exam strategy

• Write powers of ten separately to avoid mistakes.
• Always check the coefficient range at the end.
• Show working clearly: coefficient step, index step, final adjustment.

Summary

To multiply or divide numbers in standard form, handle the coefficients and powers of ten separately. Multiply/divide the coefficients, add/subtract the indices, and adjust the result back into standard form. This method simplifies working with extremely large or small numbers.