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How to Convert Scientific Notation to Standard Notation

Now we will take a closer look at how to convert from scientific notation back to standard notations. The main idea is to rewrite the power of 10, and then multiply the number $a$ by this number.

Rule

Scientific Notation to Standard Notation

Calculate the power of 10 (ex. $1 0^{4} = 10 000$ ).
Multiply the two numbers together.

Rule

Moving Comma – Scientific Notation to Standard Notation

For large numbers, when $n > 1$ in $1 0^{n}$ , you need to move the comma to the right to write it as a standard notation. The number will then be greater than $1$ .
For small numbers, when $n < 0$ in $1 0^{n}$ , you need to move the comma to the left to write it as a standard notation. The number is then between $0$ and $1$ .

Here are some examples where the power of 10 has a positive exponent. Keep in mind that the exponent is telling us how many decimal places to move the comma to the right. Note that it is the opposite way of what you did when you went from standard notation to scientific notation.

Example 1

Write $1.253 \cdot 1 0^{9}$ in standard notation.

\begin{array}{l} 1.253 \cdot 1 0^{9} & = 1.253 \cdot 1 000 000 000 \\ = 1 253 000 000 \end{array}

1.253 \cdot 1 0^{9} = 1.253 \cdot 1 000 000 000 = 1 253 000 000

Example 2

Write $6.24 \cdot 1 0^{5}$ in standard notation.

\begin{array}{l} 6.24 \cdot 1 0^{5} & = 6.24 \cdot 100 000 \\ = 624 000 \end{array}

6.24 \cdot 1 0^{5} = 6.24 \cdot 100 000 = 624 000

In the examples below, you can see how to proceed when a number in scientific notation has a negative exponent. You can imagine that the exponent tells us how many zeros there should be in total, or how many places you need to move the comma to the left. Note that it is the opposite of what you did when you went from standard notation to scientific notation.

Example 3

Write $6.24 \cdot 1 0^{- 5}$ as a standard notation.

\begin{array}{l} 6.24 \cdot 1 0^{- 5} & = 6.24 \cdot 0.000 01 \\ = 0.000 062 4 \end{array}

6.24 \cdot 1 0^{- 5} = 6.24 \cdot 0.000 01 = 0.000 062 4

Example 4

Write $1.982 \cdot 1 0^{- 3}$ in standard notation.

\begin{array}{l} 1.982 \cdot 1 0^{- 3} & = 1.982 \cdot 0.001 \\ = 0.001 982 \end{array}

1.982 \cdot 1 0^{- 3} = 1.982 \cdot 0.001 = 0.001 982

Example 5

Write $9.003 \cdot 1 0^{- 6}$ in standard notation.

\begin{array}{l} 9.003 \cdot 1 0^{- 6} & = 9.003 \cdot 0.000 001 \\ = 0.000 009 003 \end{array}

9.003 \cdot 1 0^{- 6} = 9.003 \cdot 0.000 001 = 0.000 009 003

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