Calculating standard form without a calculator
Adding and subtracting
When adding and subtracting standard formA system in which numbers are written as a number greater than 1 and less than 10 multiplied by a power of 10 which may be positive or negative. numbers you have 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 \((8.5 \times 10^7) - (1.23 \times 10^4)\)
\(= 85,000,000 - 12,300\)
\(= 84,987,700\)
\(= 8.49877 \times 10^7\)
Multiplying and dividing
When multiplying and dividing you can use index laws which are applied to the powerA number that is multiplied by itself one or more times is raised to a power.:
- multiply or divide the first numbers
- apply the index laws to the powers
Example one
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 exponentThe power that a number may be raised to.:
- \(10^3 \times 10^9 = 10^{3 + 9} = 10^{12}\)
- \((3 \times 10^3) \times (3 \times 10^9) = 9 \times 10^{12}\)
Take care that the answer is in standard form. It is common to have to re-adjust the answer.
Example two
Calculate \((4 \times 10^9) \times (7 \times 10^{-3})\)
Multiply the first numbers \(4 \times 7 = 28\)
Apply the index law on the exponents:
- \(10^9 \times 10^{-3} = 10^{9 + -3} = 10^6\)
- \((4 \times 10^9) \times (7 \times 10^{-3}) = 28 \times 10^6\)
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 identical 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 \((2 \times 10^7) \div (8 \times 10^2)\)
- Divide the first numbers: \(2 \div 8 = 0.25\)
- Apply the index law on the exponents.
- \(10^7 \div 10^2 = 10^{7 - 2} = 10^5\)
- So \((2 \times 10^7) \div (8 \times 10^2) = 0.25 \times 10^5\)
But \(0.25 \times 10^5\) is not in standard form as the first number is not between 1 and 10. To correct this, multiply 0.25 by 10 so that it is a number between 1 and 10. To balance out that multiplication of 10, divide the second part by 10 which gives 104. So \(0.25 \times 10^5\) and \(2.5 \times 10^4\) are identical but only the second is written in standard form.
So \((2 \times 10^7) \div (8 \times 10^2) = 2.5 \times 10^4\)