In short

An exponent is shorthand for repeated multiplication: a^n means a multiplied by itself n times. From that single definition, six laws follow, and they all turn out to be the same one law in different costumes — adding exponents corresponds to multiplying powers, and multiplying exponents corresponds to applying a power to a power. The laws force the strange-looking definitions a^0 = 1 and a^{-n} = \dfrac{1}{a^n}, not as a convention but as the only choices that keep the laws consistent. Once you trust the laws, you can write numbers like 7.2 \times 10^{22} for the molecules in a glass of water, or compare 2^{100} to 3^{50} at a glance, without ever computing them in full.

You probably already know that 2^{10} = 1024. The question is whether you know what's coming next. 2^{20} is about a million. 2^{30} is about a billion. 2^{40} is about a trillion. 2^{64} is the number you would get on the last square of a chessboard if you put one grain of rice on the first square and doubled at every step — and the answer is around 1.8 \times 10^{19}, more grains of rice than have ever been grown in the entire history of agriculture. The doubling went very slowly for the first few squares, and then it ran away.

That runaway behaviour is the whole reason exponents need their own notation. Writing 2^{64} as 2 \times 2 \times 2 \times \dots would take you 64 multiplications and an arm cramp; writing it as 2^{64} takes four characters. And once you have the notation, the laws of exponents let you manipulate enormous numbers without ever expanding them — you can ask "is 2^{100} bigger than 3^{50}?" and answer it in two lines, even though both numbers have more than 20 digits.

This article is the operating manual for exponents: what they mean, the small set of laws they obey, the surprising-but-forced definitions for a^0 and a^{-n}, the scientific-notation system that astronomers and biologists actually use to write very large and very small numbers, and the trick for comparing exponential numbers without computing them.

What an exponent means

An exponent is a tally that says how many times to multiply a number by itself. The notation is

a^n = \underbrace{a \times a \times a \times \dots \times a}_{n \text{ copies of } a}

The bottom number a is called the base; the top number n is called the exponent or index. Together, a^n is read "a to the power n," or "a raised to the n." In Indian textbooks the n is often called the index, and you may see a^n described as "a raised to the index n." Same thing.

Three concrete cases:

2^3 = 2 \times 2 \times 2 = 8
5^4 = 5 \times 5 \times 5 \times 5 = 625
10^6 = 10 \times 10 \times 10 \times 10 \times 10 \times 10 = 1{,}000{,}000

Powers of 10 are the most useful family for everyday life, because in our base-ten system they are just "1 followed by n zeros." That property is what makes scientific notation possible later in this article.

A small but important detail: a^1 means "a multiplied by itself one time," which is just a. So 7^1 = 7, \pi^1 = \pi, and a^1 = a for every a. The "one time" base case is uninteresting but it is needed to make the laws below come out clean.

The figure below plots the value of 2^n for n from 0 to about 6. Drag the red point along the curve to see how the value blows up: each step to the right doubles the height. By n = 6, the value is 64 — already off the top of any reasonable graph.

Interactive plot of two to the power n with a draggable point on the curveA coordinate plane showing the curve of two raised to n for n from zero to about six. The vertical axis is labelled with values up to eighty. The curve starts at one when n is zero and rises smoothly, doubling between each integer value of n. Integer points one, two, four, eight, sixteen, thirty-two, and sixty-four are marked. A draggable red point sits on the curve and a readout shows the current values of n and two to the n.n2ⁿ123456163264↔ drag the red point
The curve $y = 2^n$. Each integer marker on the curve doubles the previous one — $1, 2, 4, 8, 16, 32, 64$ — and the values run off the top of the graph by $n = 7$. Drag the red point to see the value at any $n$, including non-integer $n$. The curve does not have flat portions or jumps; it is continuous, and the laws of exponents below let you make sense of values like $2^{2.5}$ or $2^{-1}$.

The laws of exponents

There are six laws. They look different on the page, but they are all the same idea — exponents count multiplications, so the laws of exponents are the laws of counting multiplications. Each one has a one-line proof from the definition.

1. Product law: a^m \times a^n = a^{m+n}

When you multiply two powers of the same base, you can add the exponents. Why? Because

a^m \times a^n = \underbrace{(a \times \dots \times a)}_{m \text{ copies}} \times \underbrace{(a \times \dots \times a)}_{n \text{ copies}} = \underbrace{a \times \dots \times a}_{m + n \text{ copies}} = a^{m+n}

You took m copies of a and stuck them together with n more copies of a. The total number of copies is m + n. The "gluing together" step uses associativity of multiplication, which is one of the structural rules from Operations and Properties.

Concrete example. 2^3 \times 2^4 = 8 \times 16 = 128, and 2^{3 + 4} = 2^7 = 128. They match.

2. Quotient law: \dfrac{a^m}{a^n} = a^{m - n}

When you divide two powers of the same base, you can subtract the exponents. This is the same idea as the product law, run in reverse. From the definition:

\frac{a^m}{a^n} = \frac{\overbrace{a \times a \times \dots \times a}^{m \text{ copies}}}{\underbrace{a \times a \times \dots \times a}_{n \text{ copies}}}

Cancel n copies of a from the top and bottom (which leaves m - n copies on top, assuming m > n). The result is a^{m - n}.

Concrete example. \dfrac{3^5}{3^2} = \dfrac{243}{9} = 27, and 3^{5 - 2} = 3^3 = 27.

3. Power of a power: \left(a^m\right)^n = a^{m \cdot n}

When you raise a power to another power, multiply the exponents. The proof uses the product law:

\left(a^m\right)^n = \underbrace{a^m \times a^m \times \dots \times a^m}_{n \text{ copies}}

Each copy of a^m is m copies of a. There are n copies of a^m. Total copies of a: m \times n = mn. So \left(a^m\right)^n = a^{mn}.

Concrete example. \left(2^3\right)^4 = 8^4 = 4096, and 2^{3 \times 4} = 2^{12} = 4096.

4. Power of a product: (a \cdot b)^n = a^n \cdot b^n

A product raised to a power can be split into the product of the two powers. From the definition:

(ab)^n = \underbrace{(ab)(ab)(ab) \dots (ab)}_{n \text{ copies}}

Now use commutativity and associativity to regroup: pull all the as to the left and all the bs to the right. You get \underbrace{(a \times \dots \times a)}_{n} \times \underbrace{(b \times \dots \times b)}_{n} = a^n \cdot b^n.

Concrete example. (2 \times 5)^3 = 10^3 = 1000, and 2^3 \times 5^3 = 8 \times 125 = 1000.

5. Power of a quotient: \left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}

A fraction raised to a power can be split into the powers of the top and bottom. The proof is the same as the previous law, applied to the fraction:

\left(\frac{a}{b}\right)^n = \underbrace{\frac{a}{b} \times \frac{a}{b} \times \dots \times \frac{a}{b}}_{n \text{ copies}} = \frac{a^n}{b^n}

using the multiplication-of-fractions rule from Fractions and Decimals.

Concrete example. \left(\dfrac{2}{3}\right)^3 = \dfrac{8}{27}, and indeed \dfrac{2^3}{3^3} = \dfrac{8}{27}.

6. Same base, same exponent: a^n = b^n \Rightarrow a = b (when a, b > 0)

If two positive powers with the same exponent are equal, then their bases are equal too. This is the rule that lets you "take the n-th root" of both sides of an equation. It is the inverse of the laws above.

The product law is the most important of the six, because the other laws can all be derived from it. If you remember just one, remember this one: adding exponents corresponds to multiplying powers.

Visual proof of the product law a cubed times a squared equals a to the fifthTwo stacks of identical cubes representing a cubed and a squared. The first stack has three cubes labelled a a a. The second stack has two cubes labelled a a. An arrow joins them into a single longer stack of five cubes labelled a a a a a, which is a to the fifth.aaaa³ (3 copies)×aaa² (2 copies)=aaaaaa⁵ (5 copies)3 + 2 = 5
The product law as a counting argument. Three copies of $a$, multiplied with two more copies of $a$, gives five copies of $a$ in total. The exponents on the left ($3$ and $2$) add up to the exponent on the right ($5$). This is the entire content of $a^m \times a^n = a^{m + n}$ — the addition is happening on the *count*, not on the values themselves.

Zero and negative exponents

So far the exponent has been a positive integer — the "number of times you multiply a by itself." That definition does not say what a^0 should mean (you cannot multiply a by itself zero times), and it certainly does not say what a^{-3} should mean. But you can force a definition by demanding that the laws above continue to hold even in those cases.

Why a^0 = 1 (for any a \neq 0)

Apply the quotient law to \dfrac{a^n}{a^n}:

\frac{a^n}{a^n} = a^{n - n} = a^0

But the left side is also just 1, because any nonzero number divided by itself is 1. So a^0 = 1, if the quotient law is going to hold for m = n.

There is no other choice. If you defined a^0 to be anything else — 0, say, or a — the quotient law would break for the case m = n. The definition a^0 = 1 is forced by consistency with the laws.

(The case a = 0 is special: 0^0 is undefined, because the laws give contradictory answers. By the argument above 0^0 would equal 1, but by another natural argument — "0^n = 0 for any positive n, so the limit should be 0" — it would equal 0. The two arguments disagree, so 0^0 is left without a value in standard arithmetic.)

Why a^{-n} = \dfrac{1}{a^n}

Apply the quotient law to \dfrac{a^0}{a^n}:

\frac{a^0}{a^n} = a^{0 - n} = a^{-n}

But the left side is \dfrac{1}{a^n}, since a^0 = 1. So a^{-n} = \dfrac{1}{a^n}, if the quotient law and the rule a^0 = 1 are both going to hold.

Again, there is no other choice. The definition a^{-n} = \tfrac{1}{a^n} is the only one that lets the laws of exponents survive into the negative-exponent territory.

This is one of the most important moves in mathematics: defining new things by demanding that old laws continue to work. It looks like a sleight of hand the first time you see it — "you just defined a^0 to be 1!" — but it is actually the most principled possible move. The new definition isn't arbitrary; it is the only definition that doesn't break what you already had. The same trick will reappear in Roots and Radicals, where fractional exponents like a^{1/2} are defined by demanding that the power-of-power law continues to hold — and the answer turns out to be a^{1/2} = \sqrt{a}.

Scientific notation

Exponents make it possible to write very large and very small numbers in a compact, comparable form. The convention is:

\text{(a number between 1 and 10)} \times 10^{\text{(some integer)}}

For example, the speed of light in a vacuum is approximately 3 \times 10^8 metres per second — that is, 300{,}000{,}000 m/s, or 3 followed by eight zeros. The mass of an electron is about 9.11 \times 10^{-31} kilograms — that is, a decimal point followed by 30 zeros and then 911. Both of these numbers are essentially impossible to read in their fully-written-out forms, and impossible to compare to each other. In scientific notation they are easy to read and easy to compare.

The rules for converting:

The "nice form" requirement is that the leading number is between 1 and 10 (not including 10). This ensures every number has a unique scientific-notation form. Without that requirement you could write 300{,}000{,}000 as 30 \times 10^7 or 0.3 \times 10^9, both of which are correct but ambiguous.

A few real-world numbers in scientific notation:

You can multiply two numbers in scientific notation directly using the laws of exponents. For example, (3 \times 10^8) \times (2 \times 10^4) = (3 \times 2) \times 10^{8 + 4} = 6 \times 10^{12}. You multiply the leading numbers normally, and you add the exponents. The product law is doing all the work.

Comparing numbers in exponential form

Which is larger, 2^{100} or 3^{50}? Both numbers have more than 20 digits, so writing them out is unhelpful. But the laws of exponents let you compare them in two lines.

The trick is to rewrite both with the same exponent. Use the power-of-a-power law in reverse:

2^{100} = (2^2)^{50} = 4^{50}
3^{50} = 3^{50}

Now both numbers are raised to the same exponent (50), so you only need to compare the bases. 4 > 3, so 4^{50} > 3^{50}, and therefore 2^{100} > 3^{50}. The exponential notation has done all the heavy lifting; you never had to write down a single digit of either number.

The same trick works in the other direction. To compare numbers with the same base, just compare the exponents: 5^{17} < 5^{20} because 17 < 20, no calculation needed. To compare numbers with neither the same base nor the same exponent, get them into a common form first — common base, or common exponent, whichever is easier — and then read off the answer.

Two worked examples

Example 1: Simplify $\dfrac{(2^3 \times 5^2)^2}{2 \times 5^3}$

Three different exponent laws will get used here, and naming each one as it gets applied is the cleanest way to work through it.

Step 1. Apply the power-of-a-product law to the numerator's bracket.

(2^3 \times 5^2)^2 = (2^3)^2 \times (5^2)^2

Why: when a product is raised to a power, you can split the power across each factor. This breaks the bracket into two simpler powers.

Step 2. Apply the power-of-a-power law to each piece.

(2^3)^2 = 2^{3 \cdot 2} = 2^6
(5^2)^2 = 5^{2 \cdot 2} = 5^4

So the numerator is 2^6 \times 5^4.

Step 3. Rewrite the whole expression.

\frac{2^6 \times 5^4}{2^1 \times 5^3}

Why: the denominator's 2 is just 2^1 — it helps to make the index explicit before the next step.

Step 4. Apply the quotient law to each base.

\frac{2^6}{2^1} = 2^{6 - 1} = 2^5
\frac{5^4}{5^3} = 5^{4 - 3} = 5^1

Result. \dfrac{(2^3 \times 5^2)^2}{2 \times 5^3} = 2^5 \times 5 = 32 \times 5 = 160.

Simplification of an exponent expression in four labelled stepsFour boxed expressions stacked vertically connected by labelled arrows. The first box contains the original expression. An arrow labelled power of a product leads to a box where the bracket has been split. An arrow labelled power of a power leads to a box with the simplified numerator. A final arrow labelled quotient law leads to the simplified result two to the fifth times five equals one hundred sixty.(2³ × 5²)² / (2 × 5³)power of a product(2³)² × (5²)² / (2 × 5³)power of a power2⁶ × 5⁴ / (2 × 5³)quotient law2⁵ × 5 = 160
Each step replaces the previous expression with an equivalent one until the answer is just an integer. The same three laws — power-of-a-product, power-of-a-power, and quotient — handle every case where you need to reduce a heavy-looking exponent expression.

Example 2: Compare $4^{20}$ and $8^{15}$

Both numbers are huge, and writing them out is hopeless. The trick is to find a common base.

Step 1. Look for a common base that both 4 and 8 are powers of.

Both 4 and 8 are powers of 2: 4 = 2^2 and 8 = 2^3. So you can rewrite both numbers with 2 as the base.

Why: once both numbers share a base, you only have to compare exponents, which is trivial.

Step 2. Apply the power-of-a-power law to each.

4^{20} = (2^2)^{20} = 2^{2 \cdot 20} = 2^{40}
8^{15} = (2^3)^{15} = 2^{3 \cdot 15} = 2^{45}

Step 3. Compare the exponents.

The first number is 2^{40}, the second is 2^{45}. Since 40 < 45 and the bases are equal positive numbers, 2^{40} < 2^{45}.

Result. 4^{20} < 8^{15}. The second number is larger — and in fact, it is 2^{45 - 40} = 2^5 = 32 times larger.

Comparing four to the twenty and eight to the fifteen by rewriting both as powers of twoTwo horizontal arrows pointing from the left side to the right. The top arrow starts at four to the twenty and ends at two to the forty after rewriting four as two squared. The bottom arrow starts at eight to the fifteen and ends at two to the forty-five after rewriting eight as two cubed. A vertical comparison bracket joins the two right-hand sides and indicates that two to the forty is less than two to the forty-five.4²⁰4 = 2²2⁴⁰8¹⁵8 = 2³2⁴⁵<
Both numbers were rewritten as powers of $2$ — the only base that they share — and then the exponents could be compared at a glance. $40 < 45$, so $4^{20} < 8^{15}$. The original numbers each had more than ten digits, but you never needed to look at any of them.

Common confusions

Going deeper

If you came here for the laws of exponents and a working understanding of how to compute with them, you have it. The rest of this section is for readers who want to see how exponents extend beyond integers, and the strange but principled story of fractional and irrational powers.

Fractional exponents

So far the exponent has been an integer (positive, zero, or negative). But the laws extend cleanly to fractional exponents, and again the extension is forced by consistency.

What should a^{1/2} mean? Apply the power-of-a-power law:

\left(a^{1/2}\right)^2 = a^{(1/2) \cdot 2} = a^1 = a

So a^{1/2} is a number whose square is a. The only such number (for positive a) is \sqrt{a}, the square root. Therefore a^{1/2} = \sqrt{a} — forced by the laws of exponents, not chosen as a convention.

The same argument gives a^{1/n} = \sqrt[n]{a} for any positive integer n, and combining with the integer-exponent laws gives a^{m/n} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m. This is the entire content of Roots and Radicals — roots are just exponents in disguise.

Irrational exponents

What about a^{\sqrt{2}}? The "count multiplications" definition has long since stopped making sense — you cannot multiply a by itself \sqrt{2} times. But the curve y = a^x from the figure at the top of this article is continuous, with no holes or jumps, and you can extend the definition of a^x to all real x by demanding that the curve stay continuous.

The technical version of "demanding continuity" is to define a^x as a limit of a^{r} for rational numbers r approaching x. This is the kind of construction you meet properly in a chapter on limits — and it is the same construction that underlies Number Systems' real-line picture, where every irrational number is approached by rationals from both sides.

The upshot: a^x is well-defined for every real exponent x, positive integer or fractional or irrational, and the laws of exponents continue to hold for all of them. The exponential function a^x is one of the most important continuous functions in all of mathematics, and it shows up everywhere from compound interest to radioactive decay to wave equations.

Why scientific notation works

Scientific notation works because the laws of exponents reduce the multiplication of two large numbers to two simpler operations: multiplying small numbers, and adding integers. When you compute (3 \times 10^8) \times (4 \times 10^{12}), the answer is 12 \times 10^{20}, which you tidy up to 1.2 \times 10^{21}. The hard part — multiplying two huge numbers — has been replaced by 3 \times 4 = 12 and 8 + 12 = 20, both of which a child can do.

This is why physics and astronomy and chemistry use scientific notation universally: it converts the awkward arithmetic of huge numbers into the easy arithmetic of small numbers plus integer exponents. The laws of exponents are doing the lifting; the notation is just making them visible.

Where this leads next

Exponents are the bridge between arithmetic and the kind of mathematics that handles growth, decay, and very large or very small quantities.