GCSE Maths Practice: standard-form

Question 10 of 10

Multiply two numbers in standard form and present the result in correct standard form.

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\( \begin{array}{l}\text{Write } (6.2 \times 10^5)(3.1 \times 10^3) \\ \text{in standard form.}\end{array} \)

Choose one option:

Do coefficients first, exponents second, then check the coefficient range before finalising.

Multiplying Numbers in Standard Form

Standard form (scientific notation) writes numbers as \( a \times 10^n \) where \(1 \le a < 10\) and \(n\) is an integer. Multiplying in this format is efficient because coefficients and powers are handled separately, following the index laws. This skill is central to GCSE Maths Higher problems, especially when quantities span many orders of magnitude.

Core Rule

For \((a\times10^m)(b\times10^n)\), multiply coefficients and add exponents:

\[ (a\times10^m)(b\times10^n)=(ab)\times10^{m+n}. \]

If \(ab\) is not between 1 and 10, renormalise (shift the decimal) and adjust the power of ten accordingly so the result returns to true standard form.

Step-by-Step Method

  1. Multiply coefficients: compute \(ab\).
  2. Add exponents: calculate \(m+n\).
  3. Renormalise: if \(ab\ge10\) (or \(<1\)), move the decimal and modify the power of ten so \(1\le a<10\).
  4. State the final answer in \(a\times10^k\) form with appropriate significant figures, if required.

Worked Example 1 (Different Numbers)

Compute \((4.8\times10^4)(2.5\times10^2)\).

Multiply coefficients: \(4.8\times2.5=12.0\). Add exponents: \(4+2=6\). So we have \(12.0\times10^6\). Renormalise: \(12.0=1.20\times10^1\), giving \(1.20\times10^{7}\).

Worked Example 2 (Small and Large)

Compute \((7.5\times10^{-3})(3.2\times10^{6})\).

Coefficients: \(7.5\times3.2=24.0\). Exponents: \(-3+6=3\). We get \(24.0\times10^3\). Renormalise: \(24.0=2.40\times10^1\) so the answer is \(2.40\times10^{4}\).

Why Renormalising Matters

Standard form requires the coefficient to be between 1 and 10. If the raw product’s coefficient is outside this interval, your answer is not in correct standard form. Renormalising prevents presentation errors and is a common exam check.

Common Mistakes

  • Adding coefficients to exponents: Only powers are added; coefficients are multiplied.
  • Forgetting to renormalise: An answer like \(19.6\times10^5\) must be rewritten as \(1.96\times10^6\).
  • Arithmetic slips with coefficients: Use a calculator or estimate to sanity-check (e.g., \(6\times3\approx18\)).
  • Dropping significant figures carelessly: If a question specifies s.f., round at the end, not mid-calculation.

Real-World Connections

Standard form multiplication appears in physics (force and distance products), chemistry (concentration and volume), computing (data rate and time), and finance (price and quantity at scale). It enables clear communication across magnitudes without writing many zeros.

Practice Checklist

  • Separate coefficient multiplication from exponent addition.
  • Write an intermediate line before renormalising.
  • Ensure the final coefficient is \(1\le a<10\).
  • Apply rounding only if requested (e.g., to 2 or 3 s.f.).

FAQ

Q1: What if one coefficient is a whole number and the other is in standard form?
A: Convert the whole number if helpful (e.g., \(20=2.0\times10^1\)), or multiply directly then renormalise.

Q2: Can I renormalise in more than one step?
A: Yes, but one clean step is preferred: move the decimal to make \(1\le a<10\) and adjust the exponent by the same number of places.

Q3: How do I check quickly?
A: Estimate: round coefficients (e.g., \(6.2\approx6\), \(3.1\approx3\)) so \(6\times3\approx18\) and powers add to \(10^{8}\); expect about \(1.8\times10^9\). This validates order of magnitude.

Study Tip

Write two short lines: one for coefficient multiplication and exponent addition, one for renormalising. This structure earns method marks and reduces careless errors on GCSE Higher problems.