Operations with Standard Form (Cambridge International IGCSE Mathematics)

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 box x 10 ^ x button to enter numbers in standard form

  • e.g. $( 3 \times 10^{8 ) \times ( 2 \times 10^{-3} } )$
  • 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 $( 3 \times 10^{2} ) \times ( 4 \times 10^{5} )$

• STEP 1 Multiply the ordinary numbers

  • e.g. $3\times4=12$

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

  • e.g. $10^{2}\times10^{5}=10^{2+5}=10^{7}$

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

  • e.g. $12\times10^{7}$

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

  • e.g. $12=1.2\times10^{1}$ and $1.2\times10^{1}\times10^{7}=1.2\times10^{8}$

Division

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

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

  • e.g. $2\div8=0.25$

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

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

    • Be careful with negatives!

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

  • e.g. $0.25\times10^{-2}$

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

  • e.g. $0.25=2.5\times10^{-1}$ and $2.5\times10^{-1}\times10^{-2}=2.5\times10^{-3}$

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

Method 1

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

• STEP 1 Convert both numbers to ordinary numbers

  • e.g. $3.2\times10^{3}=3200$ and $2.1\times10^{2}=210$

• 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. $3410=3.41\times10^{3}$

Method 2

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

• 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. $4\times10^{50}=4\times10^{2}\times10^{48}=400\times10^{48}$

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

    • 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. $398=3.98\times10^{2}$ and $3.98\times10^{2}\times10^{48}=3.98\times10^{50}$

• This method works for negative powers too

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

    • $8\times10^{-20}=8\times10^{1}\times10^{-21}=80\times10^{-21}$
    • $( 80 \times 10^{- 21} ) - ( 5 \times 10^{- 21} ) = 75 \times 10^{- 21}$
    • $75=7.5\times10^{1}$ and $7.5\times10^{1}\times10^{-21}=7.5\times10^{-20}$

Exam Hint The second method is the most efficient when the powers of 10 have large positive or large negative indices.

Worked Example Without using a calculator, find $( 45 \times 10^{- 3} ) \div ( 0.9 \times 10^{5} )$.

Write your answer in the form $A\times10^{n}$, where $1\leq A<10$ and $n$ is an integer.

Answer:

Rewrite the division as a fraction, then separate out the powers of 10

$\frac{45\times10^{-3}}{0.9\times10^{5}}=\frac{45}{0.9}\times\frac{10^{-3}}{10^{5}}$

Work out $\frac{45}{0.9}$

$\frac{45}{0.9}=\frac{450}{9}=50$

Work out $\frac{10^{-3}}{10^{5}}$ using laws of indices

$\frac{10^{-3}}{10^{5}}=10^{-3-5}=10^{-8}$

Combine back together

equation$( 45 \times 10^{- 3} ) \div ( 0.9 \times 10^{5 ) = 50 \times 10^{-8} }$

Rewrite in standard form, where $a$ is between 1 and 10

$50\times10^{-8}=5\times10\times10^{-8}=5\times10^{-7}$

$5\times10^{-7}$

Worked Example Without using a calculator, find $( 2.8 \times 10^{- 6} ) + ( 9.7 \times 10^{- 8} )$.

Write your answer in the form $A\times10^{n}$, where $1\leq A<10$ and $n$ is an integer.

Answer:

Rewrite $2.8\times10^{-6}$ so that the power of 10 matches $9.7\times10^{-8}$

$2.8\times10^{-6}=280\times10^{-8}$

The numbers can now be added together, keeping the power of 10 the same

$( 280 \times 10^{- 8} ) + ( 9.7 \times 10^{- 8} ) = 289.7 \times 10^{- 8}$

Write the number in standard form

$2.897\times10^{2}\times10^{-8}=2.897\times10^{2-8}$

$2.897\times10^{-6}$