Performing calculations with very big or small numbers can be difficult. Such calculations, for example those related to space, can be made easier by converting numbers in and out of standard form.
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It is useful to look at patterns to try to understand negative indices:
\(3 \times 10^4 = 3 \times 10 \times 10 \times 10 \times 10 = 30,000\)
\(3 \times 10^3 = 3 \times 10 \times 10 \times 10 = 3,000\)
\(3 \times 10^2 = 3 \times 10 \times 10 = 300\)
\(3 \times 10^1 = 3 \times 10 = 30\)
\(3 \times 10^0 = 3 \times 1 = 3\) (because \(10^0 = 1\))
\(3 \times 10^{-1} = 3 \times 0.1 = 0.3\)
\(3 \times 10^{-2} = 3 \times 0.1 \times 0.1 = 0.03\)
\(3 \times 10^{-3} = 3 \times 0.1 \times 0.1 \times 0.1 = 0.003\)
Write 0.0005 in standard form.
0.0005 can be written as \(5 \times 0.0001\)
\(0.0001 = 10^{-4}\)
So \(0.0005 = 5 \times 10^{-4}\)
What is 0.000009 in standard form?
0.000009 can be written as \(9 \times 0.000001\)
\(0.000001 = 10^{-6}\)
So \(0.000009 = 9 \times 10^{-6}\)
This process can also be sped up by considering where the first digit is compared to the units column.
0.03 = \(3 \times 10^{-2}\) because the 3 is 2 places away from the units column.
0.000039 = \(3.9 \times 10^{-5}\) because the 3 is 5 places away from the units column.
What is 0.000059 in standard form?
\(5.9 \times 10^{-5}\) because the 5 is 5 places away from the units column.