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MathidiaMathematics Academy
Indices and standard form
Lesson 9 of 75
Challenge

Calculate in standard form

Objective. Multiply and divide numbers written in standard form.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

To calculate with numbers in standard form, treat the coefficients and powers of 10 separately. For multiplication, multiply the coefficients together and add the powers of 10; for division, divide the coefficients and subtract the powers. After combining, check whether the resulting coefficient is still between 1 and 10, adjusting the power of 10 if it is not. For example, (3 × 10⁵) × (2 × 10³) = 6 × 10⁸, since 3 × 2 = 6 and 5 + 3 = 8. If the coefficients multiply to give 10 or more, such as 4 × 5 = 20, rewrite 20 × 10ⁿ as 2 × 10ⁿ⁺¹.

Worked example

Calculate (6 × 10⁷) × (5 × 10⁻³), giving your answer in standard form.

Array of 6 rows by 10 columns
  1. 1Multiply the coefficients: 6 × 5 = 30.
  2. 2Add the powers of 10: 7 + (−3) = 4.
  3. 3This gives 30 × 10⁴, but 30 is not between 1 and 10.
  4. 4Rewrite 30 × 10⁴ as 3 × 10⁵.

Answer: 3 × 10⁵

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Common mistake. Pupils often forget to renormalise when the coefficients multiply beyond 10, leaving an answer like 24 × 10⁶ instead of converting it to 2.4 × 10⁷. Always check the coefficient is between 1 and 10 at the end.

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