Calculating with Standard Form

Grade 6

The key idea: handle the parts separately

When you calculate with numbers in standard form, deal with the a part and the power of 10 separately. This keeps things tidy and avoids errors.

Multiplying and dividing are straightforward. Adding and subtracting have one extra step — you need both numbers written with the same power of 10 first.

Multiplying and dividing

To multiply: multiply the a values together, then add the powers of 10. To divide: divide the a values, then subtract the powers. After either operation, check that your result is still in standard form — if the a part has come out below 1 or equal to 10 or greater, adjust it.

EXAMPLE
Calculate (4 × 10⁵) × (3 × 10⁴). Give your answer in standard form.
Multiply the a values: 4 × 3 = 12. Add the powers: 5 + 4 = 9. This gives 12 × 10⁹. But 12 is not between 1 and 10, so rewrite it: 12 = 1.2 × 10¹. Combine: 1.2 × 10¹ × 10⁹ = 1.2 × 10¹⁰
EXAMPLE
Calculate (9 × 10⁶) ÷ (3 × 10²). Give your answer in standard form.
Divide the a values: 9 ÷ 3 = 3. Subtract the powers: 6 − 2 = 4. The a value 3 is already between 1 and 10, so no adjustment needed. 3 × 10⁴

Adding and subtracting

You can only add or subtract standard form numbers directly when their powers match. If they don't, convert one of them so both share the same power of 10, then add or subtract the a values.

It's usually easier to convert to the larger power. That keeps numbers manageable — though either way works as long as the powers match.
EXAMPLE
Calculate (5.2 × 10⁶) + (3 × 10⁵). Give your answer in standard form.
The powers differ (6 and 5). Rewrite the second number with power 6: 3 × 10⁵ = 0.3 × 10⁶. Now add the a values: 5.2 + 0.3 = 5.5. The power stays at 6. 5.5 × 10⁶ ✓

Watch out after every calculation

Always check your final a value is between 1 and 10. Multiplying two a values can give you something ≥ 10, and adding them can do the same. If it's not in range, rewrite it — one extra step, but it's required for the mark.
Nova
Try this one

Put these numbers in order of size, starting with the smallest.

\[ 5.1 \times 10^3, \quad 4.8 \times 10^4, \quad 9.2 \times 10^2, \quad 6.3 \times 10^3 \]

[2]

Nova's hint:

This uses the technique of ordering numbers given in standard form by comparing their powers first, then their values of \( a \).

Ask yourself: 'Which numbers have the same power of 10, and how do I compare those?' Try converting each to an ordinary number first if it helps.

Try to Solve it with Nova?