Calculating standard form without a calculator

Adding and subtracting

Explore updated revision resources for GCSE Maths: Standard form, with step-by-step slideshows, quizzes, practice exam questions, and more!

When adding and subtracting numbers, an easy way is to:

convert the numbers from standard form into decimal form or ordinary numbers
complete the calculation
convert the number back into standard form

Example

Calculate \((4.5 \times 10^4) + (6.45 \times 10^6)\).

\(= 45,000 + 6,450,000\)

\(= 6,495,000\)

\(= 6.495 \times 10^6\)

Question

Calculate \((8.5 \times 10^7) - (1.23 \times 10^4)\).

Multiplying and dividing

When multiplying and dividing you can use the Laws of Indices:

multiply or divide the first numbers
apply the Laws of Indices to the powers of 10

Example 1

Calculate \((3 \times 10^3) \times (3 \times 10^9)\).

Multiply the first numbers – which in this case is \(3 \times 3 = 9\).

Apply the index law on the powers of 10:

\(10^3 \times 10^9 = 10^{3 + 9} = 10^{12}\)
\((3 \times 10^3) \times (3 \times 10^9) = 9 \times 10^{12}\)

Take care that the answer is in standard form. It is common to have to re-adjust the answer.

Example 2

Calculate \((4 \times 10^9) \times (7 \times 10^{-3})\).

Multiply the first numbers \(4 \times 7 = 28\).

Apply the Laws of Indices on the powers:

\(10^9 \times 10^{-3} = 10^{9 + -3} = 10^6\)
\((4 \times 10^9) \times (7 \times 10^{-3}) = 28 \times 10^6\)

But \(28 \times 10^6\) is not in standard form, as the first number is not between 1 and 10. To correct this, divide 28 by 10 so that it is a number between 1 and 10. To balance out that division of 10, multiply the second part by 10 which gives 10⁷.

\(28 \times 10^6\) and \(2.8 \times 10^7\) are equivalent but only the second is written in standard form.

So: \((4 \times 10^9) \times (7 \times 10^{-3}) = 2.8 \times 10^7\)

Question

Calculate \((2 \times 10^7) \div (8 \times 10^2)\).

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Standard form - OCRCalculating standard form without a calculator