How to Use Exponents to Write Powers of Ten?

Powers of ten are the foundation of our base-10 number system and are essential for understanding place value, scientific notation, and large numbers. Because 10 is our base, multiplying by powers of ten simply shifts the decimal point — and exponents give us a concise way to write those powers. This lesson shows you how.

What Are Powers of Ten?

A power of ten is the result of multiplying 10 by itself one or more times. Using exponents:

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• \(\color{blue}{10^{0} = 1}\)
• \(\color{blue}{10^{1} = 10}\)
• \(\color{blue}{10^{2} = 100}\)
• \(\color{blue}{10^{3} = 1,000}\)
• \(\color{blue}{10^{4} = 10,000}\)
• \(\color{blue}{10^{5} = 100,000}\)
• \(\color{blue}{10^{6} = 1,000,000}\)

The exponent equals the number of zeros in the standard form — or equivalently, the number of times you move the decimal point to the right.

How to Use Exponents to Write Powers of Ten

From standard form to exponential form

Count the zeros after the 1. That count is the exponent.

• \(\color{blue}{100}\) has 2 zeros → \(\color{blue}{10^{2}}\)
• \(\color{blue}{1,000,000}\) has 6 zeros → \(\color{blue}{10^{6}}\)

From exponential form to standard form

Write a 1 followed by (exponent) zeros.

• \(\color{blue}{10^{4}}\) → write 1 followed by 4 zeros: \(\color{blue}{10,000}\)

Multiplying by a power of ten

Multiplying a number by \(\color{blue}{10^{n}}\) moves the decimal point n places to the right (or equivalently, adds n zeros if the number is a whole number).

• \(\color{blue}{47 \times 10^{3} = 47,000}\)
• \(\color{blue}{3.5 \times 10^{2} = 350}\)

Step-by-Step Summary

1. To write a power of ten in exponential form: count the zeros in the standard number; that count is the exponent.
2. To expand a power of ten: write 1 followed by (exponent) zeros.
3. To multiply by a power of ten: shift the decimal (exponent) places to the right.
4. To divide by a power of ten: shift the decimal (exponent) places to the left.

Watch: Powers of Ten (Video Lesson)

Math with Mr. J explains the powers-of-ten pattern and how to use them in the base-10 place-value system:

Worked Examples

Example 1: Write 100,000 as a power of ten.

Count the zeros: 5 zeros after the 1. So \(\color{blue}{100,000 = 10^{5}}\).
Answer: 10⁵

Example 2: Expand \(\color{blue}{10^{4}}\).

Write 1 followed by 4 zeros: \(\color{blue}{10,000}\).
Answer: 10,000

Example 3: Evaluate \(\color{blue}{35 \times 10^{3}}\).

Move decimal 3 places right: \(\color{blue}{35 \times 1,000 = 35,000}\).
Answer: 35,000

Example 4: A computer processes \(\color{blue}{10^{6}}\) instructions per second. Write this as a standard number.

\(\color{blue}{10^{6} = 1,000,000}\).
Answer: 1,000,000 instructions per second

More Practice: Multiplying Whole Numbers by Powers of Ten

Math with Mr. J shows how multiplying by powers of ten works with whole numbers and place value:

Exercises

1. Write 1,000,000,000 as a power of ten.
2. Expand \(\color{blue}{10^{7}}\) into standard form.
3. Calculate \(\color{blue}{42 \times 10^{4}}\).
4. A city has a population of \(\color{blue}{10^{5}}\) people. Write the population as a standard number.
5. Write each as a power of ten: 1; 10; 1,000; 10,000,000.

Answers

1. \(\color{blue}{10^{9}}\)
2. \(\color{blue}{10,000,000}\)
3. \(\color{blue}{420,000}\)
4. \(\color{blue}{100,000 \text{ people }}\)
5. \(\color{blue}{10^{0}; 10^{1}; 10^{3}; 10^{7}}\)

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Frequently Asked Questions

Why does 10⁰ = 1?

Any nonzero number raised to the power of 0 equals 1. You can see this pattern in the sequence: 10³ = 1,000; 10² = 100; 10¹ = 10. Each step divides by 10, so 10⁰ = 10 ÷ \(\color{blue}{10 = 1}\).

How are powers of ten used in scientific notation?

Scientific notation writes a number as a value between 1 and 10 multiplied by a power of ten. For example, 4,\(\color{blue}{500 = 4.5 \times 10}\)³. Powers of ten allow scientists to write very large or very small numbers compactly.

What happens when you divide by a power of ten?

Division shifts the decimal point to the left. For example, \(\color{blue}{6,000 &\text{ div }; 10^{2} = 60}\) (decimal moves 2 places left). This is the same as multiplying by a negative power of ten.

Related Topics

• Use Exponents to Write Multiplication Expressions
• Prime Factorization with Exponents
• Evaluate Integers Raised to Rational Exponents
• Multiplying and Dividing Decimals
• Order of Operations