Calculations Using Scientific Notation (SQA National 5 Maths): Revision Note

Calculations using scientific notation

How do I perform calculations in scientific notation using a calculator?

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

  • 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 scientific notation, but the question requires it:

  • Either rewrite it using the standard process

    • e.g. 3 820 000 = 3.82 × 10⁶

  • Or rewrite numbers in scientific notation, then apply the laws of indices

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

How do I perform calculations with numbers in scientific notation without a calculator?

Multiplication and division

• Consider the "number parts" separately to the powers of 10

  • E.g. $\left(3\times {10}^2\right)\times \left(4\times {10}^5\right)$

    • Can be written as $\left(3\times 4\right)\times \left({10}^2\times {10}^5\right)$

  • Then calculate each part separately

    • Use laws of indices when combining the powers of 10

    • $12\times {10}^7$

  • This can then be rewritten in scientific notation

    • $1.2\times 10\times {10}^7$$=1.2\times {10}^8$

• This process is the same for division

  • E.g. $\left(8\times {10}^{-5}\right)÷\left(2\times {10}^{-3}\right)$

    • Can be written as $\frac{8\times {10}^{-5}}{2\times {10}^{-3}}=\frac{8}{2}\times \frac{{10}^{-5}}{{10}^{-3}}$

  • Then calculate each part separately

    • Use laws of indices when combining the powers of 10

    • Be careful with negative powers -5 -(-3) is -5 + 3

    • $4\times {10}^{-2}$

Addition and subtraction

• One strategy is to write both numbers in full, rather than scientific notation, and then add or subtract them

  • E.g. $\left(3.2\times {10}^3\right)+\left(2.1\times {10}^2\right)$

  • Can be written as $3200+\hspace{0.17em}210=3410$

  • Then this can be rewritten in scientific notation if needed, $3.41\times {10}^3$

• However this method is not efficient for very large or very small powers

• For very large or very small powers:

  • Write the values with the same, highest, power of 10

  • And then calculate the addition or subtraction, keeping the power of 10 the same

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

    • Rewrite both with the highest power of 10, i.e. 50

    • Changing 10⁴⁸ to 10⁵⁰ has made it 10² times larger, so make the 2 smaller by a factor of 10² to compensate

    • $\left(4\times {10}^{50}\right)+\left(0.02\times {10}^{50}\right)$

    • These can now be added

    • $4.02\times {10}^{50}$

  • Consider $\left(8\times {10}^{-20}\right)-\left(5\times {10}^{-21}\right)$

    • Rewrite both with the higher power of 10, i.e. -20

    • Changing 10^-21 to 10^-20 has made it 10¹ times larger, so make the five 10¹ times smaller to compensate

    • $\left(8\times {10}^{-20}\right)-\left(0.5\times {10}^{-20}\right)$

    • These can now be subtracted

    • $7.5\times {10}^{-20}$