Converting scientific notation to engineering notation, review exercises on powers and roots, and giving a result to the appropriate number of significant figures when the data comes from measurement.

Engineering notation is scientific notation with one extra condition: the exponent must be a multiple of three. That way the power matches a prefix - kilo, mega, milli, micro - and the result reads straight off in units. The second part of the lesson is about accuracy: if the data comes from measurement, the result cannot be more precise than the data.

Engineering notation

1) Convert the numbers in scientific notation to engineering notation

\text{a)}\quad 8.236 \cdot 10^{7} = 82.36 \cdot 10^{-1} \cdot 10^{7} = 82.36 \cdot 10^{6}

\text{b)}\quad 1.624 \cdot 10^{-4} = 162.4 \cdot 10^{-2} \cdot 10^{-4} = 162.4 \cdot 10^{-6}

\text{c)}\quad 4.827 \cdot 10^{4} = 48.27 \cdot 10^{-1} \cdot 10^{4} = 48.27 \cdot 10^{3}

\text{d)}\quad 6.243 \cdot 10^{5} = 624.3 \cdot 10^{-2} \cdot 10^{5} = 624.3 \cdot 10^{3}

\text{e)}\quad 3.274 \cdot 10^{-2} = 32.74 \cdot 10^{-1} \cdot 10^{-2} = 32.74 \cdot 10^{-3}

\text{f)}\quad 5.362 \cdot 10^{-7} = 536.2 \cdot 10^{-2} \cdot 10^{-7} = 536.2 \cdot 10^{-9}

In engineering notation, the exponent is restricted to a multiple of 3, for example 10³, 10⁶, 10⁻³, 10⁻⁶ …

2) (4.72 · 10²) · (8.36 · 10⁵)

\text{a)}\quad 4.72 \cdot 8.36 \cdot 10^{7} = 39.4592 \cdot 10^{7} = 3.94592 \cdot 10^{8}

scientific notation

\text{b)}\quad 394.592 \cdot 10^{-2} \cdot 10^{8} = 394.592 \cdot 10^{6}

engineering notation

Review exercises

Write each of the following numbers as a power

\text{a)}\quad 5^{8} \cdot 5^{2} = 5^{10}

\text{b)}\quad 6^{4} : 6^{6} = 6^{-2}

\text{c)}\quad (7^{4})^{3} = 7^{12}

\text{d)}\quad (19^{-8})^{0} = 19^{0} = 1

Find the value of each of the following numbers, correct to three decimal places

\text{a)}\quad 16^{\tfrac{1}{4}} = \pm 2

\text{b)}\quad \sqrt[3]{3} = 3^{\tfrac{1}{3}} = 1.442

\text{c)}\quad (-8)^{\tfrac{1}{5}} = -1.516

\text{d)}\quad (-7)^{\tfrac{1}{4}}

does not exist

Write each of the following expressions as a single decimal number

\text{a)}\quad 1.0521 \cdot 10^{3} = 1052.1

\text{b)}\quad 0.0135 : 10^{-3} = 0.0135 \cdot 10^{3} = 13.5

\text{c)}\quad 123.456 \cdot 10^{-2} = 1.23456

\text{d)}\quad 165.21 : 10^{4} = 165.21 \cdot 10^{-4} = 0.016521

Write each of the following numbers in scientific notation

\text{a)}\quad 125.87 = 1.2587 \cdot 10^{2}

\text{b)}\quad 0.0101 = 1.01 \cdot 10^{-2}

\text{c)}\quad 1.345 = 1.345 \cdot 10^{0}

\text{d)}\quad 10.13 = 1.013 \cdot 10^{1}

Write each of the following numbers in engineering notation

\text{a)}\quad 1.3204 \cdot 10^{5} = 132.04 \cdot 10^{-2} \cdot 10^{5} = 132.04 \cdot 10^{3}

\text{b)}\quad 0.0101 = 10.1 \cdot 10^{-3}

\text{c)}\quad 1.345 = 1.345 \cdot 10^{0}

\text{d)}\quad 9.5032 \cdot 10^{-8} = 95.032 \cdot 10^{-1} \cdot 10^{-8} = 95.032 \cdot 10^{-9}

In each of the following cases, the numbers were obtained by measurement. Give the result of each calculation to the appropriate degree of accuracy

\text{a)}\quad 13.6 : 0.012 \cdot 7.63 - 9015 = -367{,}(6) = -370

to two significant figures

\text{b)}\quad \frac{0.003 \cdot 194}{13.6} = 0.042794 = 0.04

to one significant figure

\text{c)}\quad 19.3 \cdot 1.04^{2.00} = 20.87488 = 20.9

to three significant figures

\text{d)}\quad \frac{18 \cdot 2.1 - 3.6 \cdot 0.54}{8.6 \cdot 2.9 + 5.7 \cdot 9.2} = 0.46338 = 0.46

to two significant figures

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Frequently asked questions

How does engineering notation differ from scientific notation?

In scientific notation the mantissa lies between one and ten and the exponent can be anything. In engineering notation the exponent must be a multiple of three, so the mantissa can be larger - for example 8.236 times ten to the seventh is written as 82.36 times ten to the sixth.

What are significant figures and why limit a result to them?

They are the digits that carry information about the value, counted from the first non-zero digit. When data comes from measurement the result cannot be more precise than the measurement - so 0.042794 is given as 0.04 if the data had one significant figure.

Why does the fourth root of a negative number not exist?

Because the degree is even, and any real number raised to an even power is non-negative. With an odd degree there is no problem - the fifth root of minus eight is about minus 1.516.

What is a number raised to the power of zero?

Always one, no matter what the base is. That is why nineteen to the power of minus eight, all raised to the power of zero, gives one.

A lesson from the PhiBoard knowledge base: phiboard.com/en/knowledge-base/from-zero-to-engineer/engineering-notation. On the lesson page you will also find games and practice exercises - free and without an account.

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Roots, fractional exponents and scientific notation

Number systems: binary, octal, duodecimal and hexadecimal