How to Use Exponents to Write Powers of Ten?

How to use exponents to write powers of ten, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.

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:

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

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

How do you write a number in expanded form using exponents?

Write each digit times its place value, then write each place value as a power of ten. For 4,305: 4 × 10³ + 3 × 10² + 0 × 10¹ + 5 × 10⁰, usually shortened to 4 × 10³ + 3 × 10² + 5 × 10⁰. Since 10⁰ = 1, the ones digit is simply multiplied by 1.

What are numbers expressed using exponents called?

A number written with a base and an exponent is called a power, and that way of writing it is called exponential form. So 10⁴ is a power of ten, read as ten to the fourth power, and its standard form is 10,000. The exponent 4 tells you 10 is used as a factor 4 times, which also matches the number of zeros after the 1.

How do you write an expression using exponents?

Count how many times the same factor repeats and use that count as the exponent. For powers of ten, 10 × 10 × 10 × 10 × 10 = 10⁵. Starting from a standard number, count the zeros after the 1: 1,000,000 has 6 zeros, so it equals 10⁶. A number like 3,000 can be written as 3 × 10³.

How do the laws of exponents work with powers of ten?

The usual exponent rules apply, and they make powers of ten quick to handle. When multiplying, add the exponents: 10³ × 10⁴ = 10⁷. When dividing, subtract them: 10⁸ ÷ 10⁵ = 10³. A power of a power multiplies them: (10²)³ = 10⁶. These rules are the basis of scientific notation, which you will also see on GED Math.

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