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Year 10 Maths | Understand, justify, check

Use powers of ten to make scale explicit

Scientific notation separates a number’s significant digits from its scale. It makes very large and very small quantities easier to compare without counting long strings of zeros. The exponent belongs to the value, not to the number of digits in the coefficient.

What to notice

Standard scientific notation uses a coefficient at least 1 and less than 10. A positive exponent scales up; a negative exponent scales down. Both parts must be read together.

What to do

Write the power of ten as a scale factor, then multiply. When comparing two standard-form values with different exponents, compare those powers before comparing the coefficients.

Why it works

A shared representation makes orders of magnitude visible. It also provides a check on place-value errors: multiplying by ten changes the scale by one power, not by an arbitrary shift of zeros.

Work through an example

Write 4.2 × 10⁵ as an ordinary number.

  1. 10⁵ = 100000.
  2. 4.2 × 100000 = 420000.
  3. The result lies between 10⁵ and 10⁶, consistent with a standard-form coefficient of 4.2.

A common mistake

A negative exponent does not make the number negative. It gives a reciprocal power of ten: 10⁻³ is 0.001, which is positive.

A strategy to try

Estimate the interval containing the answer before converting. Then count place-value steps and check whether a small quantity has remained below one.

Try it now

Try three questions

No account is needed. These examples do not change saved learning progress. Choose an answer and check it for an explanation; you can try again.

1. Write 6.3 × 10⁴ as an ordinary number.
Help: a starting point

Write the power of ten as an ordinary number.

Help: the next step

Multiply the coefficient by that scale factor, checking the place values.

2. Write 4.2 × 10⁻³ as a decimal.
Help: a starting point

A negative exponent represents a reciprocal power of ten.

Help: the next step

For power −3, divide the coefficient by one thousand.

3. Which value is greater: 8.1 × 10⁵ or 1.2 × 10⁶?
Help: a starting point

Both coefficients are in standard scientific-notation range.

Help: the next step

Compare their powers of ten, or convert both to ordinary numbers.

For parents and carers

Ask the learner to explain the exponent as a scale factor. These examples cover conversion and comparison, not significant-figure rounding or uncertainty analysis. Avoid treating every zero as significant without information about how a measurement was recorded.

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