The Sun is about 93 million miles away, but scientists rarely write out 93,000,000. Instead they write 9.3 x 10^7. Why? Because exponents let you pack long strings of zeros into a tiny, tidy form. Once you know what an exponent means and the few rules that go with it, scientific notation stops looking like code and starts looking like a shortcut you can use every day.
An exponent tells you how many times to multiply a number by itself. In a^n, the base is a and the exponent is n, so 3^4 = 3 x 3 x 3 x 3 = 81. Scientific notation writes a number as m x 10^n, where m is between 1 and 10 and n moves the decimal point. It keeps very big and very small numbers short, like 9.3 x 10^7 for 93 million.
What an Exponent Is
An exponent is a small number that means repeated multiplication. In the expression a^n, the number a is the base and n is the exponent. The exponent tells you how many copies of the base to multiply together.
For example, 3^4 means 3 x 3 x 3 x 3. You multiply four 3s to get 81. The base is the thing being multiplied, and the exponent counts how many times it appears.
We read a^2 as “a squared” and a^3 as “a cubed.” Everything else is read as “to the power of,” so 2^5 is “2 to the power of 5.” Exponents are just a compact way to write multiplication you would otherwise do the long way.
The Main Exponent Rules
A handful of rules cover almost every exponent problem you will meet. Each one is a shortcut that saves you from writing out long multiplication. Here are the five most useful ones.
- Product rule: a^m x a^n = a^(m+n). When you multiply powers with the same base, add the exponents.
- Quotient rule: a^m / a^n = a^(m-n). When you divide powers with the same base, subtract the exponents.
- Power rule: (a^m)^n = a^(m*n). When you raise a power to another power, multiply the exponents.
- Zero rule: a^0 = 1 for any nonzero base a. Any number to the power of 0 equals 1.
- Negative rule: a^-n = 1/a^n. A negative exponent means one divided by the positive power.
These rules only work when the bases match. If the bases are different, you cannot simply add or subtract the exponents.
Worked Examples With Exponents
Rules make more sense with numbers. Here is one worked example for each rule, with every step shown.
Repeated multiplication: 3^4 = 3 x 3 x 3 x 3. Multiply left to right: 3 x 3 = 9, then 9 x 3 = 27, then 27 x 3 = 81. So 3^4 = 81.
Product rule: 2^3 x 2^4. Same base, so add the exponents: 3 + 4 = 7. That gives 2^7 = 128. Check it: 8 x 16 = 128.
Quotient rule: 5^6 / 5^2. Same base, so subtract the exponents: 6 – 2 = 4. That gives 5^4 = 625. The shared base lets you shorten a big division into one small power.
Power rule: (2^3)^2. Multiply the exponents: 3 x 2 = 6, so the answer is 2^6 = 64. Check it: 8 squared is 64.
Zero rule: 5^0 = 1. Any nonzero number raised to the power of 0 is 1.
Negative rule: 2^-3 = 1/2^3 = 1/8 = 0.125. The negative sign flips the power into a fraction.
If you want to check bigger powers fast, the Exponent Calculator does the arithmetic for you and confirms your work.
What Scientific Notation Is
Scientific notation is a way to write very big or very small numbers in a short form. Every number is written as m x 10^n. The letter m is a number that is at least 1 but less than 10, and n is a whole-number power of 10.
The power of 10 does the heavy lifting. A positive n means a large number, and it tells you how many places the decimal point moves to the right. A negative n means a small number, and it moves the decimal point to the left.
For example, 9.3 x 10^7 has m = 9.3 and n = 7. The 10^7 shifts the decimal seven places right, turning 9.3 into 93,000,000. The single value of n replaces a long trail of zeros.
Write Big and Small Numbers
To turn a plain number into scientific notation, put the decimal point right after the first nonzero digit. Then count how many places you moved it. That count becomes n.
Big number: 5,600,000. Move the decimal so you get 5.6. You moved it 6 places left, so n = 6. The answer is 5.6 x 10^6.
Another big one: 93,000,000 becomes 9.3, and the decimal moved 7 places left. So it is 9.3 x 10^7, the Sun distance from the intro.
Small number: 0.00042. Move the decimal to get 4.2. You moved it 4 places right, so n is negative: n = -4. The answer is 4.2 x 10^-4.
To go back to a plain number, just reverse the move. A positive n shifts the decimal right, and a negative n shifts it left. Learning powers of ten also helps when you convert between number systems, which our sibling guide on converting between number bases covers. If you are studying algebra more broadly, see how exponents appear in our guide on how to solve a quadratic equation.
Common Mistakes to Avoid
Exponents follow simple rules, but a few slips catch people out again and again. Watching for these will save you from wrong answers.
- Multiplying the base by the exponent: 3^4 is not 3 x 4 = 12. It is 3 x 3 x 3 x 3 = 81. The exponent counts copies of the base, not a factor.
- Thinking a negative exponent gives a negative answer: 2^-3 is 1/8, a positive fraction, not -8.
- Combining different bases: you cannot add exponents in 2^3 x 5^2, because the bases differ.
- Forgetting the range of m: in scientific notation, m must be at least 1 and less than 10. Writing 56 x 10^5 is not proper form; it should be 5.6 x 10^6.
- Moving the decimal the wrong way: a positive power makes a number bigger, and a negative power makes it smaller.
When unsure, write the multiplication out the long way and count carefully. A quick check by hand catches most errors.
Ready to test your own powers and check every step? Try our Exponent Calculator. Enter any base and exponent, including negative and zero powers, and see the exact result instantly so you can confirm your math with confidence.
Frequently Asked Questions About Exponents and Scientific Notation
What Is an Exponent in Simple Terms?
An exponent is a small number that tells you how many times to multiply a base by itself. In a^n, a is the base and n is the exponent. For example, 3^4 means 3 x 3 x 3 x 3, which equals 81. It is just a short way to write repeated multiplication.
What Does a Zero Exponent Mean?
Any nonzero number raised to the power of 0 equals 1. So 5^0 = 1 and 100^0 = 1. This follows from the quotient rule: a^n / a^n equals a^(n-n), which is a^0, and any number divided by itself is 1.
How Do Negative Exponents Work?
A negative exponent means one divided by the positive power. So a^-n = 1/a^n. For example, 2^-3 = 1/2^3 = 1/8 = 0.125. The negative sign does not make the answer negative; it makes it a fraction instead.
What Is Scientific Notation Used For?
Scientific notation writes very big or very small numbers in a short form, m x 10^n, where m is between 1 and 10. It keeps long strings of zeros manageable. Scientists use it for things like distances in space and the size of atoms, and it makes such numbers easier to compare.
How Do I Write a Big Number in Scientific Notation?
Place the decimal point after the first nonzero digit, then count how many places you moved it. That count is the power n. For example, 5,600,000 becomes 5.6 x 10^6 because the decimal moved 6 places left. Big numbers always use a positive power of 10.
How Do I Write a Small Number in Scientific Notation?
Move the decimal point right until it sits after the first nonzero digit, then count the moves. That count becomes a negative power. For example, 0.00042 becomes 4.2 x 10^-4 because the decimal moved 4 places right. Small numbers below 1 always use a negative power.
Can I Add Exponents With Different Bases?
No. The product rule a^m x a^n = a^(m+n) only works when the bases are the same. For different bases, like 2^3 x 5^2, you cannot combine the exponents. You have to work out each power first, so 8 x 25 = 200 in that example.
Sources
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This article is for general education only. Always double-check the formulas and math for your own problem before you rely on the result. Reviewed for accuracy by Prof. Dr. Khalil Mudassar, PhD. Last updated September 11, 2026.
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