What is an exponent, and how do the laws of exponents work?

An exponent is the small raised number that tells you how many times to multiply the base by itself, so 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. The laws of exponents follow from that: multiply powers of the same base by adding the exponents, divide by subtracting them, and raise a power to a power by multiplying them.

An exponent is the small raised number written after a base, and it tells you how many times to use the base as a factor. In 10³ the base is 10 and the exponent is 3, so 10³ = 10 × 10 × 10 = 1,000. You read it as "10 to the third power" or "10 cubed".

What does a negative exponent mean?

A negative exponent means one divided by the same power with a positive exponent. So 2⁻³ = 1⁄2³ = 1⁄8, and 10⁻² = 1⁄100 = 0.01. The answer stays positive when the base is positive, because the minus sign in the exponent points to a reciprocal.

Why is any number to the power of 0 equal to 1?

Any nonzero number to the power of 0 equals 1 because each step down in the exponent divides by the base once. For base 2, the pattern runs 2³ = 8, 2² = 4, 2¹ = 2, and one more division by 2 gives 2⁰ = 1. The quotient law agrees: 5⁴ ÷ 5⁴ = 5⁰, and any nonzero number divided by itself is 1.

Do you add exponents when multiplying?

You add exponents when you multiply two powers of the same base, so 3² × 3³ = 3⁵ = 243. The rule does not work for different bases: 2³ × 3² is 8 × 9 = 72, and the exponents cannot be combined. Adding is also wrong for a power of a power, where (2³)² = 2⁶ and the exponents multiply.

How do you type an exponent?

A calculator, spreadsheet or plain-text message writes an exponent with the caret sign, so 2^5 means 2⁵. In Excel or Google Sheets, =2^5 and =POWER(2,5) both return 32. In a word processor, type the number and format the exponent as superscript.

Exponents Explained: Rules and Worked Examples

Your child's homework asks for 2⁵, and the answer on the page says 10. Exponents look like a small detail, a number written high and small, yet that detail turns 10 into 32. One idea sits under all the laws of exponents, and once it is clear the rules stop being a list to memorize.

What is an exponent?

An exponent is the small raised number that tells you how many times to use the base as a factor. In 2⁵, the base is 2 and the exponent is 5, so 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. The whole expression is called a power, and you read it as "2 to the fifth power".

The OpenStax Prealgebra textbook puts it in one line: "the exponent tells us how many times we use the base a as a factor." Repeated multiplication is to exponents what repeated addition is to multiplication: 2 × 5 adds five 2s, and 2⁵ multiplies five 2s.

Powers of 2 grow by doubling. Each step multiplies by one more 2, so the chain runs 2, 4, 8, 16, 32, and ten steps reach 2¹⁰ = 1,024.

A doubling chain of block grids showing 2 to the first through 2 to the fifth power as 2, 4, 8, 16 and 32 blocks, with the full product 2 × 2 × 2 × 2 × 2 = 32 above and a reminder that 2 to the fifth is not 2 × 5 = 10.

Squared and cubed: where the names come from

The exponents 2 and 3 have their own names because they describe shapes. A square with sides of 4 holds 4 × 4 = 16 unit squares, so 4² is read "4 squared". A cube with edges of 3 holds 3 × 3 × 3 = 27 unit cubes, so 3³ is read "3 cubed".

England's national curriculum for mathematics sets this in Year 5, where pupils "recognise and use square numbers and cube numbers, and the notation for squared (²) and cubed (³)". The square numbers game drills the first twelve squares until 7² = 49 and 12² = 144 come back without counting.

Play free in your browserSquare NumbersGrade 4–6

Why exponents matter: powers of 10 and place value

Exponents first appear in school as powers of 10, because they describe place value. Each place in a number is worth 10 times the place to its right, so the places are 10¹ = 10, 10² = 100, 10³ = 1,000 and 10⁶ = 1,000,000.

Common Core standard 5.NBT.A.2 asks fifth graders to "explain patterns in the number of zeros of the product when multiplying a number by powers of 10" and to "use whole-number exponents to denote powers of 10". The exponent counts the zeros: 4 × 10³ = 4,000, and 3.7 × 10² = 370, where the decimal point moves two places to the right.

The guide to place value for parents covers the columns behind that pattern. Exponents also run through prime factorization, where 360 is written 2³ × 3² × 5 in the post on prime numbers explained.

Exponents in the order of operations

Exponents come second in the order of operations, after parentheses and before multiplication and division. Common Core standard 6.EE.A.1 asks sixth graders to "write and evaluate numerical expressions involving whole-number exponents", and standard 6.EE.A.2c names the conventional order.

So 3 + 2² × 5 = 3 + 4 × 5 = 23, because the exponent goes first. Parentheses change that: (3 + 2)² = 5² = 25. The full sequence is in the guide to the order of operations explained.

How to simplify an exponent expression, step by step

An exponent expression simplifies in a fixed order: check the bases, combine exponents with the right law, then evaluate. Take 2³ × 2⁴ ÷ 2⁵ as the worked example.

Check that the bases match. All three powers have base 2, so the laws apply. With different bases, such as 2³ × 3², evaluate each power on its own: 8 × 9 = 72.
Multiply by adding exponents. 2³ × 2⁴ = 2³⁺⁴ = 2⁷, because three 2s times four 2s gives seven 2s.
Divide by subtracting exponents. 2⁷ ÷ 2⁵ = 2⁷⁻⁵ = 2², because five of the seven 2s cancel.
Evaluate the power. 2² = 2 × 2 = 4.
Check by expanding. 2³ × 2⁴ = 8 × 16 = 128, and 128 ÷ 32 = 4. The two routes agree.

Step 5 is the habit worth building. A law that gives a different answer from the long way has been applied wrong.

Play free in your browserOrder of OperationsGrade 5+

Where the laws stop working

The laws of exponents only combine powers that share a base and are multiplied or divided. They do nothing for sums: 2³ + 2⁴ = 8 + 16 = 24, which is not 2⁷ = 128. A sum inside a power does not split either, so (3 + 4)² = 7² = 49, while 3² + 4² = 9 + 16 = 25.

A product inside a power does split. (2 × 5)³ = 10³ = 1,000, and 2³ × 5³ = 8 × 125 = 1,000, because multiplication can be regrouped and addition cannot.

Why anything to the power of 0 equals 1

Any nonzero number to the power of 0 equals 1, and a pattern shows why. Write the powers of 2 from the top down: 2⁴ = 16, 2³ = 8, 2² = 4, 2¹ = 2. Each step down divides by 2, so the next step gives 2 ÷ 2 = 1, and that is 2⁰.

The quotient law reaches the same place. 5⁴ ÷ 5⁴ = 5⁴⁻⁴ = 5⁰, and any nonzero number divided by itself is 1. OpenStax states the definition: "If a is a non-zero number, then a⁰ = 1."

The halving pattern for powers of 2: 2 to the fourth is 16, then 8, 4, 2, then 2 to the zero is 1, then 2 to the minus one is one half and 2 to the minus two is one quarter, with a divide-by-2 arrow between each card.

Negative exponents: a grade-8 preview

Negative exponents continue the same pattern past zero. One more division by 2 gives 2⁻¹ = 1⁄2, and the next gives 2⁻² = 1⁄4, so a negative exponent means a reciprocal. In general, 2⁻³ = 1⁄2³ = 1⁄8, and 10⁻² = 1⁄100 = 0.01.

Common Core standard 8.EE.A.1 places this in grade 8, with the example 3² × 3⁻⁵ = 3⁻³ = 1⁄3³ = 1⁄27. A fifth or sixth grader does not need the rule yet, and seeing the pattern now means the rule arrives as something familiar.

The laws of exponents side by side

The laws of exponents are five rules, and each one is a shortcut for writing out the repeated multiplication. The comparison puts each law next to a worked example and the wrong answer students most often write.

Law	Rule	Worked example	Common wrong answer
Product law	aᵐ × aⁿ = aᵐ⁺ⁿ	3² × 3³ = 3⁵ = 243	3⁶ = 729 (exponents multiplied)
Quotient law	aᵐ ÷ aⁿ = aᵐ⁻ⁿ	5⁶ ÷ 5⁴ = 5² = 25	1² = 1 (bases divided)
Power of a power	(aᵐ)ⁿ = aᵐⁿ	(2³)² = 2⁶ = 64	2⁵ = 32 (exponents added)
Zero exponent	a⁰ = 1, a ≠ 0	7⁰ = 1	0
Negative exponent	a⁻ⁿ = 1/aⁿ	2⁻³ = 1/8	−8

The OpenStax power property says it in words: "To raise a power to a power, multiply the exponents." The product law is the one that adds, and mixing those two laws up accounts for two of the five wrong answers in the comparison.

How to rebuild a law you forgot

A forgotten law of exponents can be rebuilt in under a minute by expanding a small example. For (2³)², write 2³ twice: (2 × 2 × 2) × (2 × 2 × 2). That is six 2s, so the exponents multiply, 3 × 2 = 6.

For 2³ × 2², write (2 × 2 × 2) × (2 × 2). That is five 2s, so the exponents add, 3 + 2 = 5. Small bases and small exponents keep the check short, and the method works for every law in the comparison.

How to check that exponents have landed

Five signals show whether exponents make sense to your child or to you, and each takes under a minute.

The meaning, said aloud: Ask what 4³ means. "4 × 4 × 4, which is 64" is the answer; "4 × 3" is not.
Powers of 10 by sight: Ask for 10⁴ and 6 × 10². The answers are 10,000 and 600.
The order of operations: Give 2 × 3². The answer is 18, because 3² = 9 comes first; 36 means the multiplication went first.
A law, with a check: Give 2⁴ × 2³. The answer 2⁷ = 128 should come with the check 16 × 8 = 128.
The zero exponent, with a reason: Ask why 9⁰ = 1. A pattern or the quotient law is a reason; "it's the rule" is a memory.

A five-minute test

A five-minute test mixes one item of each type: a power to evaluate, a power of 10, an order-of-operations line, one law and one zero exponent. Repeat it with new numbers twice a week for two weeks.

Mark the reason as well as the answer. A child who writes 2⁵ = 32 and can show the five 2s has the idea; a child who writes 32 because it looks familiar can drift back to 10 a month later. The math game for 6th graders page gathers practice at the grade where exponents enter the Common Core.

Common exponent mistakes and edge cases

Five mistakes cause most wrong answers with exponents, and each has a one-line fix.

2³ = 6: The exponent was used as a multiplier. Write out the factors, 2 × 2 × 2 = 8, until the meaning sticks.
−3² = 9: Without parentheses, the exponent applies to the 3 only, so −3² = −(3 × 3) = −9, while (−3)² = (−3) × (−3) = 9.
2³ × 3² = 6⁵: The bases were combined. Different bases are evaluated separately, so the answer is 8 × 9 = 72, while 6⁵ = 7,776.
(a + b)² = a² + b²: A sum does not split. Test it with (3 + 4)² = 49 against 3² + 4² = 25.
2⁻³ = −8: A negative exponent gives a reciprocal, so 2⁻³ = 1⁄8, which is positive.

Three edge cases are worth knowing. The number 1 to any power is 1, so 1¹⁰⁰ = 1, and 0 to any positive power is 0, so 0⁵ = 0. School textbooks leave 0⁰ out, which is why the zero-exponent rule reads "a ≠ 0".

Your exponents checklist

Say the meaning: the exponent counts how many times the base is a factor.
Count zeros: 10ⁿ is 1 followed by n zeros.
Keep the order: exponents come after parentheses and before multiplication.
Match bases first: the product and quotient laws need the same base.
Check the long way: expand a small example when a law feels uncertain.

Adults returning to algebra meet exponents again in scientific notation and compound interest, and the guide on how to get better at math as an adult covers the routine that rebuilds fluency. The math game for adults page covers practice built for grown-up learners.

MathIt is a free app for iOS and Android with about 90 game types and adaptive difficulty, so square numbers and order-of-operations practice get harder only after the easier items come out right. Twenty-two of its games play free in a browser with no sign-up.