How to Calculate with Scientific Notation

Here, you will take a look at how you do calculations with numbers in scientific notations. It’s a piece of cake. The main idea is to work separately with the power of 10’s and the numbers in front of the powers.

Rule

Multiplication with Scientific Notations

\begin{array}{l} (a \cdot 1 0^{n}) \cdot (b \cdot 1 0^{m}) & = a \cdot 1 0^{n} \cdot b \cdot 1 0^{m} \\ = a \cdot b \cdot 1 0^{n + m} \end{array}

(a \cdot 1 0^{n}) \cdot (b \cdot 1 0^{m}) = a \cdot 1 0^{n} \cdot b \cdot 1 0^{m} = a \cdot b \cdot 1 0^{n + m}

Rule

Division with Scientific Notation

\frac{a \cdot 1 0^{n}}{b \cdot 1 0^{m}} = \frac{a}{b} \cdot \frac{1 0^{n}}{1 0^{m}} = \frac{a}{b} \cdot 1 0^{n - m}

Example 1

Calculate $(4 \cdot 1 0^{4}) \cdot (2 \cdot 1 0^{2})$

You use the multiplication rule.

\begin{array}{l} (4 \cdot 1 0^{4}) \cdot (2 \cdot 1 0^{2}) & = 4 \cdot 2 \cdot 1 0^{4 + 2} \\ = 8 \cdot 1 0^{6} \end{array}

(4 \cdot 1 0^{4}) \cdot (2 \cdot 1 0^{2}) = 4 \cdot 2 \cdot 1 0^{4 + 2} = 8 \cdot 1 0^{6}

Example 2

Calculate $5 \cdot 1 0^{4} \cdot 3 \cdot 1 0^{5}$

Remember to sort.

\begin{array}{l} 5 \cdot 1 0^{4} \cdot 3 \cdot 1 0^{5} & = 5 \cdot 3 \cdot 1 0^{4 + 5} \\ = 15 \cdot 1 0^{9} \\ = 1.5 \cdot 10 \cdot 1 0^{9} \\ = 1.5 \cdot 1 0^{9 + 1} \\ = 1.5 \cdot 1 0^{10} \end{array}

A value in scientific notation has a number from $1$ to $10$ multiplied by a power of 10. Since $5 \cdot 3 =$ 15, you will need to convert $15$ to a number between $1$ and $10$ . To do this, write $15 = 1.5 \cdot 10$ (in scientific notations) and repeat once more.

Example 3

Calculate $\frac{8 \cdot 1 0^{6}}{4 \cdot 1 0^{4}}$

You use the division rule.

\begin{array}{l} \frac{8 \cdot 1 0^{6}}{4 \cdot 1 0^{4}} & = \frac{8}{4} \cdot \frac{1 0^{6}}{1 0^{4}} = 2 \cdot 1 0^{6 - 4} \\ = 2 \cdot 1 0^{2} = 200 \end{array}

\frac{8 \cdot 1 0^{6}}{4 \cdot 1 0^{4}} = \frac{8}{4} \cdot \frac{1 0^{6}}{1 0^{4}} = 2 \cdot 1 0^{6 - 4} = 2 \cdot 1 0^{2} = 200

Example 4

Calculate $\frac{3 \cdot 1 0^{9}}{6 \cdot 1 0^{5}}$ and write in scientific notation

You factorize into two fractions

\begin{array}{l} \frac{3 \cdot 1 0^{9}}{6 \cdot 1 0^{5}} & = \frac{3}{6} \cdot \frac{1 0^{9}}{1 0^{5}} \\ = 0.5 \cdot 1 0^{9 - 5} \\ = 0.5 \cdot 1 0^{4} \\ = 5 \cdot 1 0^{- 1} \cdot 1 0^{4} \\ = 5 \cdot 1 0^{3} \end{array}

A value in scientific notation has a number from $1$ to $10$ multiplied by a power of 10. You must convert $0.5$ to a number between $1$ and $10$ . To do this, write $0.5 = 5 \cdot 1 0^{- 1}$ (in scientific notation) and repeat once more.

The math master thinking about scientific notation

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