How to Solve Scientific Notation? (+FREE Worksheet!)

Scientific notation is a compact way to write very large or very small numbers using powers of 10. Scientists use it to express everything from the distance to the nearest star to the size of an atom. In Algebra 1, mastering scientific notation sets you up for multiplying, dividing, adding, and subtracting such numbers quickly and accurately.

What Is Scientific Notation?

A number is in scientific notation when it is written in the form:

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\(\color{blue}{a \times 10}\)ⁿ

where \(\color{blue}{1 \le |a| < 10}\) (the coefficient has exactly one nonzero digit before the decimal point) and \(\color{blue}{n}\) is an integer.

• Large numbers: positive exponent (\(\color{blue}{4,500,000 = 4.5 \times 10^{6}}\))
• Small numbers: negative exponent (\(\color{blue}{0.00032 = 3.2 \times 10^{-4}}\))

How to Convert to and from Scientific Notation

Standard form → scientific notation

Move the decimal point until one nonzero digit remains to its left. Count the moves: moving left makes a positive exponent; moving right makes a negative exponent.

• \(\color{blue}{4,500,000}\): move decimal 6 places left → \(\color{blue}{4.5 \times 10^{6}}\)
• \(\color{blue}{0.00032}\): move decimal 4 places right → \(\color{blue}{3.2 \times 10^{-4}}\)

Scientific notation → standard form

Positive exponent: move decimal right. Negative exponent: move decimal left.

• \(\color{blue}{1.5 \times 10^{3} = 1500}\)
• \(\color{blue}{1.5 \times 10^{-3} = 0.0015}\)

Step-by-Step Summary

1. Locate the decimal point in the original number.
2. Move it until exactly one nonzero digit is to its left — that value is your coefficient \(\color{blue}{a}\).
3. Count the number of places moved; that count is the exponent \(\color{blue}{n}\).
4. Sign of \(\color{blue}{n}\): moved left → positive; moved right → negative.
5. Write \(\color{blue}{a \times 10^{n}}\) and verify \(\color{blue}{1 \le a < 10}\).

Watch: Scientific Notation (Video Lesson)

Math Antics makes converting to and from scientific notation crystal clear:

Scientific Notation – Worked Examples

Example 1: Write \(\color{blue}{4,500,000}\) in scientific notation.

Move the decimal 6 places left: \(\color{blue}{4.500000}\) → coefficient = \(\color{blue}{4.5}\), exponent = \(\color{blue}{6}\).
Answer: \(\color{blue}{4.5 \times 10^{6}}\)

Example 2: Write \(\color{blue}{0.00032}\) in scientific notation.

Move the decimal 4 places right: coefficient = \(\color{blue}{3.2}\), exponent = \(\color{blue}{-4}\).
Answer: \(\color{blue}{3.2 \times 10^{-4}}\)

Example 3: Convert \(\color{blue}{3.7 \times 10^{5}}\) to standard form.

Positive exponent: move decimal 5 places right.
Answer: \(\color{blue}{370,000}\)

Example 4: Convert \(\color{blue}{8.1 \times 10^{-3}}\) to standard form.

Negative exponent: move decimal 3 places left.
Answer: \(\color{blue}{0.0081}\)

More Practice: Intro to Scientific Notation Video

Khan Academy’s introduction covers the reasoning and additional conversion examples:

Exercises for Scientific Notation

Convert each number as indicated.

1. Write \(\color{blue}{25,000,000}\) in scientific notation.
2. Write \(\color{blue}{0.0000045}\) in scientific notation.
3. Convert \(\color{blue}{3.7 \times 10^{5}}\) to standard form.
4. Convert \(\color{blue}{8.1 \times 10^{-3}}\) to standard form.
5. Write \(\color{blue}{760,000}\) in scientific notation.
6. Write \(\color{blue}{0.000901}\) in scientific notation.

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Answers

1. \(\color{blue}{2.5 \times 10^{7}}\)
2. \(\color{blue}{4.5 \times 10^{-6}}\)
3. \(\color{blue}{370,000}\)
4. \(\color{blue}{0.0081}\)
5. \(\color{blue}{7.6 \times 10^{5}}\)
6. \(\color{blue}{9.01 \times 10^{-4}}\)

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Want More Practice?

We haven’t published a worksheet built specifically for Scientific Notation just yet. In the meantime, the free worksheets below cover closely related skills and concepts. If you’d like extra practice, download any that look helpful, complete the problems, and check your work — they’re a great way to reinforce what you learned on this page and strengthen the foundations this topic builds on:

• Download Properties of Exponents Worksheet
• Download Rational and Irrational Numbers Worksheet

Frequently Asked Questions

What makes a number correctly written in scientific notation?

The coefficient must be at least 1 and less than 10, and it must be multiplied by a power of 10. For example, \(\color{blue}{12 \times 10^{4}}\) is not correct scientific notation; the correct form is \(\color{blue}{1.2 \times 10^{5}}\).

How do I handle numbers like 10,000?

Move the decimal 4 places left: the coefficient is \(\color{blue}{1.0}\) and the exponent is \(\color{blue}{4}\), giving \(\color{blue}{1.0 \times 10^{4}}\) (often written simply as \(\color{blue}{10^{4}}\)).

Why is scientific notation useful in algebra?

It makes arithmetic with extreme numbers manageable. Multiplying and dividing in scientific notation reduces to combining coefficients and adding or subtracting exponents of 10.

Related Topics

• Addition and Subtraction in Scientific Notation
• Multiplication and Division in Scientific Notation
• Zero and Negative Exponents
• Multiplication Property of Exponents
• Division Property of Exponents