Operations with Standard Form (AQA GCSE Maths: Higher): Revision Note

Operations with standard form

How do I perform calculations in standard form using a calculator?

• Make use of brackets around each number, and use the $\boxed{\times {10}^x}$ button to enter numbers in standard form

  • e.g. $\left(3\times {10}^8\right)\times \left(2\times {10}^{-3}\right)$ 

  • You can instead use the standard multiplication and index buttons

• If your calculator answer is not in standard form, but the question requires it:

  • Either rewrite it using the standard process

    • e.g. 3 820 000 = 3.82 × 10⁶

  • Or rewrite numbers in standard form, then apply the laws of indices

    • e.g.  243 × 10²⁰ = (2.43 × 10²) × 10²⁰ = 2.43 × 10²²

How do I multiply and divide numbers in standard form without a calculator?

Multiplication

• Consider the example $\left(3\times {10}^2\right)\times \left(4\times {10}^5\right)$

• STEP 1
  Multiply the ordinary numbers

  • e.g. $3\times 4=12$

• STEP 2
  Multiply the powers of 10 by adding the indices

  • e.g. ${10}^2\times {10}^5={10}^{2+5}={10}^7$

• STEP 3
  Multiply the new ordinary number and the new power of 10

  • e.g. $12\times {10}^7$

• STEP 4 (if needed)
  Write the ordinary number in standard form and simplify the powers of 10 by adding the indices

  • e.g. $12=1.2\times {10}^1$ and $1.2\times {10}^1\times {10}^7=1.2\times {10}^8$

Division

• Consider the example $\left(2\times {10}^{-5}\right)÷\left(8\times {10}^{-3}\right)$

• STEP 1
  Divide the first ordinary number by the second ordinary number

  • e.g. $2÷8=0.25$

• STEP 2
  Divide the first power of 10 by the second power of 10 by subtracting the indices

  • e.g. ${10}^{-5}÷{10}^{-3}={10}^{-5-\left(-3\right)}={10}^{-2}$

    • Be careful with negatives!

• STEP 3
  Multiply the new ordinary number and the new power of 10

  • e.g. $0.25\times {10}^{-2}$

• STEP 4 (if needed)
  Write the ordinary number in standard form and simplify the powers of 10 by adding the indices

  • e.g. $0.25=2.5\times {10}^{-1}$ and $2.5\times {10}^{-1}\times {10}^{-2}=2.5\times {10}^{-3}$

How do I add and subtract numbers in standard form without a calculator?

Method 1

• Consider the example $\left(3.2\times {10}^3\right)+\left(2.1\times {10}^2\right)$

• STEP 1
  Convert both numbers to ordinary numbers

  • e.g. $3.2\times {10}^3=3200$ and $2.1\times {10}^2=210$

• STEP 2
  Add or subtract the ordinary numbers

  • e.g. $3200+210=3410$

• STEP 3 (if needed)
  Convert the answer to standard form

  • e.g. $3410=3.41\times {10}^3$

Method 2

• Consider the example $\left(4\times {10}^{50}\right)-\left(2\times {10}^{48}\right)$

• STEP 1
  Rewrite the number with the biggest power of 10 so that it has the same power of 10 as the number with the lowest power of 10

  • e.g. $4\times {10}^{50}=4\times {10}^2\times {10}^{48}=400\times {10}^{48}$

    • The ordinary gets bigger as the power of 10 gets smaller

• STEP 2
  Collect like terms by adding or subtracting the ordinary numbers

  • e.g. $\left(400\times {10}^{48}\right)-\left(2\times {10}^{48}\right)=398\times {10}^{48}$

    • Do not change the power of 10

• STEP 3
  Write the ordinary number in standard form and simplify the powers of 10 by adding the indices

  • e.g. $398=3.98\times {10}^2$ and $3.98\times {10}^2\times {10}^{48}=3.98\times {10}^{50}$

• This method works for negative powers too

  • e.g. consider $\left(8\times {10}^{-20}\right)-\left(5\times {10}^{-21}\right)$

    • $8\times {10}^{-20}=8\times {10}^1\times {10}^{-21}=80\times {10}^{-21}$

    • $\left(80\times {10}^{-21}\right)-\left(5\times {10}^{-21}\right)=75\times {10}^{-21}$

    • $75=7.5\times {10}^1$ and $7.5\times {10}^1\times {10}^{-21}={10}^{-20}$