Exponents

What Are Exponents?

An exponent is a shorthand way of expressing repeated multiplication. Rather than writing $5 \times 5$, we can write $5^2$. They mean the same thing—the superscript 2 tells us to multiply 5 by itself twice. Similarly, $y^4$ means multiply $y$ four times: $y \times y \times y \times y$.

It might not seem like a huge time-saver when dealing with small exponents like $5^2$, but imagine writing out $5^{25}$! That would be 25 fives multiplied together. With exponential notation, we can express this compactly.

But exponents are far more than just a convenience. They're fundamental to how we describe many relationships in math and science. The area of a square depends on the side length squared. The volume of a cube depends on the side length cubed. Population growth, radioactive decay, compound interest—all of these involve exponents. Understanding exponents opens the door to understanding how the world works.

Basic Notation

In an expression like $5^3$, we have two parts:

The base is the number being multiplied (5 in this case)
The exponent (also called the power) is the small raised number that tells us how many times to multiply (3 in this case)

So $5^3$ means $5 \times 5 \times 5 = 125$.

Example 1: Evaluate $2^4$.

This means multiply 2 by itself four times: $$2^4 = 2 \times 2 \times 2 \times 2 = 16$$

Example 2: Evaluate $3^2$.

$$3^2 = 3 \times 3 = 9$$

Example 3: Evaluate $10^3$.

$$10^3 = 10 \times 10 \times 10 = 1000$$

Notice how powers of 10 are particularly useful—each power of 10 just adds another zero.

Example 4: What about $x^5$?

If $x$ is a variable, $x^5 = x \times x \times x \times x \times x$. We can't simplify this further unless we know what $x$ equals, but we understand what it represents.

Product Rule: Multiplying Powers

Here's where exponents start to get really useful. When you multiply two powers that have the same base, you add the exponents.

$$a^m \times a^n = a^{m+n}$$

Why does this work? Let's look at a concrete example.

Example 5: Simplify $x^2 \times x^6$.

Think about what each part means:

$x^2 = x \times x$
$x^6 = x \times x \times x \times x \times x \times x$

When we multiply them together: $$x^2 \times x^6 = (x \times x) \times (x \times x \times x \times x \times x \times x)$$

We have 2 $x$'s from the first part and 6 $x$'s from the second part, giving us 8 $x$'s total: $$x^2 \times x^6 = x^8$$

Notice we added the exponents: $2 + 6 = 8$.

Example 6: Simplify $3^2 \times 3^4$.

Add the exponents: $$3^2 \times 3^4 = 3^{2+4} = 3^6 = 729$$

Example 7: Simplify $y \times y^3$.

Remember that $y$ by itself is the same as $y^1$: $$y \times y^3 = y^1 \times y^3 = y^{1+3} = y^4$$

A quick note: This rule only works when the bases are the same. You can't simplify $x^2 \times y^3$ using the product rule because $x$ and $y$ are different bases. When you see different bases, the expression stays as is.

Quotient Rule: Dividing Powers

When you divide two powers with the same base, you subtract the exponents.

$$\frac{a^m}{a^n} = a^{m-n}$$

Again, let's see why this makes sense.

Example 8: Simplify $\frac{x^7}{x^3}$.

Think of it in terms of multiplication: $$\frac{x^7}{x^3} = \frac{x \times x \times x \times x \times x \times x \times x}{x \times x \times x}$$

Three $x$'s cancel from the numerator and denominator, leaving us with four $x$'s in the numerator: $$\frac{x^7}{x^3} = x^4$$

We subtracted the exponents: $7 - 3 = 4$.

Example 9: Simplify $\frac{5^6}{5^2}$.

$$\frac{5^6}{5^2} = 5^{6-2} = 5^4 = 625$$

Example 10: Simplify $\frac{y^5}{y}$.

Remember that $y = y^1$: $$\frac{y^5}{y} = \frac{y^5}{y^1} = y^{5-1} = y^4$$

Power Rule: Raising Powers to Powers

When you raise a power to another power, you multiply the exponents.

$$(a^m)^n = a^{m \times n}$$

Example 11: Simplify $(x^2)^3$.

What does this mean? It means $x^2$ multiplied by itself three times: $$(x^2)^3 = x^2 \times x^2 \times x^2$$

Using the product rule, we add the exponents: $$x^2 \times x^2 \times x^2 = x^{2+2+2} = x^6$$

Or we could have just multiplied: $2 \times 3 = 6$, so $(x^2)^3 = x^6$.

Example 12: Simplify $(3^2)^4$.

Multiply the exponents: $$(3^2)^4 = 3^{2 \times 4} = 3^8 = 6561$$

Example 13: Simplify $(a^4)^2$.

$$(a^4)^2 = a^{4 \times 2} = a^8$$

There are also rules for raising products and quotients to powers:

Raising a product to a power: $$(ab)^n = a^n b^n$$

Example 14: Simplify $(2x)^3$.

$$(2x)^3 = 2^3 \times x^3 = 8x^3$$

Notice that both the 2 and the $x$ get cubed. The exponent applies to everything inside the parentheses. If you forgot to cube the 2, you'd get $2x^3$, which is wrong.

Raising a quotient to a power: $$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$

Example 15: Simplify $\left(\frac{x}{3}\right)^2$.

$$\left(\frac{x}{3}\right)^2 = \frac{x^2}{3^2} = \frac{x^2}{9}$$

Keep these straight: When you multiply powers with the same base like $x^2 \times x^3$, you add the exponents to get $x^5$. But when you raise a power to a power like $(x^2)^3$, you multiply the exponents to get $x^6$. Don't mix up adding and multiplying!

Zero Exponents

Here's something that surprises many students: any non-zero number raised to the power of 0 equals 1.

$$a^0 = 1 \quad \text{(as long as } a \neq 0 \text{)}$$

Why is this true? Let's use the quotient rule to see.

Consider $\frac{a^3}{a^3}$. On one hand, anything divided by itself equals 1: $$\frac{a^3}{a^3} = 1$$

On the other hand, using the quotient rule (subtract the exponents): $$\frac{a^3}{a^3} = a^{3-3} = a^0$$

Therefore, $a^0 = 1$.

Example 16: Evaluate $5^0$.

$$5^0 = 1$$

Example 17: Evaluate $(37)^0$.

$$(37)^0 = 1$$

Example 18: Simplify $3x^0$.

Be careful here! The exponent only applies to what it's directly attached to—in this case, just the $x$, not the 3: $$3x^0 = 3 \times 1 = 3$$

Example 19: Simplify $(3x)^0$.

Now the parentheses mean the exponent applies to everything inside: $$(3x)^0 = 1$$

See the difference? Where you put the parentheses matters a lot.

Negative Exponents

Negative exponents can seem intimidating at first, but there's a simple rule: a negative exponent means "take the reciprocal."

$$a^{-n} = \frac{1}{a^n}$$

In other words, to make a negative exponent positive, flip the base to the other side of a fraction bar.

Let's see why this makes sense using the quotient rule. Consider $\frac{a^2}{a^5}$:

Using the quotient rule: $\frac{a^2}{a^5} = a^{2-5} = a^{-3}$

But we can also think of it as canceling: $$\frac{a^2}{a^5} = \frac{a \times a}{a \times a \times a \times a \times a} = \frac{1}{a \times a \times a} = \frac{1}{a^3}$$

So $a^{-3} = \frac{1}{a^3}$.

Example 20: Simplify $3^{-2}$.

A negative exponent means reciprocal: $$3^{-2} = \frac{1}{3^2} = \frac{1}{9}$$

Example 21: Simplify $2^{-4}$.

$$2^{-4} = \frac{1}{2^4} = \frac{1}{16}$$

Notice the answer is positive. A negative exponent doesn't make the answer negative—it just means "reciprocal." If someone tells you $5^{-2}$ equals $-25$, they're confusing the exponent with the answer.

Example 22: Simplify $x^{-5}$.

$$x^{-5} = \frac{1}{x^5}$$

Example 23: Simplify $\frac{1}{4^{-2}}$.

If a negative exponent is in the denominator, flip it to the numerator and make it positive: $$\frac{1}{4^{-2}} = 4^2 = 16$$

Think of it this way: $4^{-2} = \frac{1}{4^2}$, so $\frac{1}{4^{-2}} = \frac{1}{\frac{1}{4^2}} = 4^2$.

Example 24: Simplify $2 \times 4^{-2}$.

$$2 \times 4^{-2} = 2 \times \frac{1}{4^2} = 2 \times \frac{1}{16} = \frac{2}{16} = \frac{1}{8}$$

Example 25: Simplify $\frac{x^{-3}}{y^{-2}}$.

Move each term with a negative exponent to the opposite part of the fraction and make the exponents positive: $$\frac{x^{-3}}{y^{-2}} = \frac{y^2}{x^3}$$

The $x^{-3}$ moved from numerator to denominator, and the $y^{-2}$ moved from denominator to numerator.

A word about final answers: Most teachers want you to express your final answer without negative exponents. So if you end up with something like $5x^{-2}$, rewrite it as $\frac{5}{x^2}$ before you call it done.

Example 26: Solve for $x$: $\frac{1}{x^{-4}} = 16$.

First, simplify the left side. Since $x^{-4}$ is in the denominator with a negative exponent, it moves to the numerator with a positive exponent: $$\frac{1}{x^{-4}} = x^4$$

So our equation becomes: $$x^4 = 16$$

Take the fourth root of both sides: $$x = \pm 2$$

We have two solutions because both $2^4 = 16$ and $(-2)^4 = 16$.

Combining Multiple Rules

Most real problems require using several exponent rules together. Let's work through some more complex examples.

Example 27: Simplify $(x^2 y^3)^2$.

Use the power rule on each factor: $$(x^2 y^3)^2 = (x^2)^2 \times (y^3)^2 = x^4 y^6$$

Example 28: Simplify $x^{-2} \times x^5$.

Use the product rule (add exponents): $$x^{-2} \times x^5 = x^{-2+5} = x^3$$

Example 29: Simplify $\frac{12x^7 y^2}{3x^3 y^5}$.

Break this into parts:

Divide the coefficients: $\frac{12}{3} = 4$
Use the quotient rule on $x$: $\frac{x^7}{x^3} = x^{7-3} = x^4$
Use the quotient rule on $y$: $\frac{y^2}{y^5} = y^{2-5} = y^{-3} = \frac{1}{y^3}$

Put it together: $$\frac{12x^7 y^2}{3x^3 y^5} = 4x^4 y^{-3} = \frac{4x^4}{y^3}$$

Example 30: Simplify $\frac{(2a^3)^2}{4a^4}$.

First, handle the numerator using the power rule: $$(2a^3)^2 = 2^2 \times (a^3)^2 = 4a^6$$

Now we have: $$\frac{4a^6}{4a^4}$$

Simplify: $$\frac{4a^6}{4a^4} = \frac{4}{4} \times \frac{a^6}{a^4} = 1 \times a^2 = a^2$$

Example 31: Simplify $\left(\frac{x^{-2}}{y^3}\right)^{-2}$.

First, deal with the negative exponent outside the parentheses. A negative exponent means flip the fraction: $$\left(\frac{x^{-2}}{y^3}\right)^{-2} = \left(\frac{y^3}{x^{-2}}\right)^2$$

Now simplify the fraction inside. The $x^{-2}$ in the denominator becomes $x^2$ in the numerator: $$\left(\frac{y^3}{x^{-2}}\right)^2 = \left(\frac{y^3 \times x^2}{1}\right)^2 = (x^2 y^3)^2$$

Finally, apply the power rule: $$(x^2 y^3)^2 = x^4 y^6$$

Real-World Applications

Exponents aren't just abstract math—they describe real phenomena all around us. Let's look at some practical applications.

Example 32: Compound Interest

If you invest $1,000 at 5% annual interest compounded yearly, after $t$ years you'll have: $$A = 1000(1.05)^t$$

After 10 years: $$A = 1000(1.05)^{10} = 1000(1.629) \approx 1,629$$

You'd have approximately $1,629. The exponent represents how many times the interest compounds.

Example 33: Population Growth

A bacteria population doubles every hour. If you start with 100 bacteria, after $h$ hours you have: $$P = 100 \times 2^h$$

After 5 hours: $$P = 100 \times 2^5 = 100 \times 32 = 3,200 \text{ bacteria}$$

Example 34: Area and Volume

If you double the side length of a square, what happens to its area?

Original square with side $s$: $A = s^2$
New square with side $2s$: $A = (2s)^2 = 4s^2$

The area quadruples! This is why larger packages cost more than just twice the price of smaller ones—they require four times the material.

For a cube, if you double the side length: Original: $V = s^3$
New: $V = (2s)^3 = 8s^3$

The volume increases by a factor of 8.

Example 35: Scientific Notation

Scientists use exponents to express very large and very small numbers:

Speed of light: $3 \times 10^8$ meters per second (300,000,000 m/s)
Mass of an electron: $9.11 \times 10^{-31}$ kilograms (0.000000000000000000000000000000911 kg)

Without exponents, these numbers would be unwieldy to write and work with.

Example 36: Half-Life

Radioactive materials decay exponentially. If a substance has a half-life of 10 years and you start with 100 grams, after $t$ years you have: $$A = 100 \times \left(\frac{1}{2}\right)^{t/10}$$

After 30 years (three half-lives): $$A = 100 \times \left(\frac{1}{2}\right)^{30/10} = 100 \times \left(\frac{1}{2}\right)^3 = 100 \times \frac{1}{8} = 12.5 \text{ grams}$$

Try These Problems

Work through these on your own, then check your answers below.

Evaluate $4^3$
Simplify $x^4 \times x^2$
Simplify $\frac{y^8}{y^3}$
Simplify $(a^3)^4$
Evaluate $7^0$
Simplify $5^{-2}$
Simplify $(2x^3)^2$
Simplify $\frac{x^{-4}}{x^{-7}}$
Simplify $\frac{18m^9 n^2}{6m^3 n^5}$
Simplify $\left(\frac{3a^2}{b^{-3}}\right)^2$

Check Your Answers

64
$4^3 = 4 \times 4 \times 4 = 64$
$x^6$
$x^4 \times x^2 = x^{4+2} = x^6$
$y^5$
$\frac{y^8}{y^3} = y^{8-3} = y^5$
$a^{12}$
$(a^3)^4 = a^{3 \times 4} = a^{12}$
1
$7^0 = 1$ (any non-zero number to the zero power equals 1)
$\frac{1}{25}$
$5^{-2} = \frac{1}{5^2} = \frac{1}{25}$
$4x^6$
$(2x^3)^2 = 2^2 \times (x^3)^2 = 4x^6$
$x^3$
$\frac{x^{-4}}{x^{-7}} = x^{-4-(-7)} = x^{-4+7} = x^3$
$\frac{3m^6}{n^3}$
$\frac{18}{6} = 3$, $\frac{m^9}{m^3} = m^6$, $\frac{n^2}{n^5} = n^{-3} = \frac{1}{n^3}$
**$\frac{9a^4 b^6}{1}$ or $9a^4 b^6$
First simplify inside: $\frac{3a^2}{b^{-3}} = 3a^2 b^3$
Then square: $(3a^2 b^3)^2 = 9a^4 b^6$

Exponents and Order of Operations

Remember PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction)? Exponents come early in the order, which means you evaluate them before multiplication, division, addition, or subtraction.

Example 37: Evaluate $2 \times 3^2$.

Do the exponent first: $$2 \times 3^2 = 2 \times 9 = 18$$

If you multiplied first, you'd get $(2 \times 3)^2 = 6^2 = 36$, which is wrong unless there were parentheses.

Example 38: Evaluate $5 + 2^3$.

Exponent first: $$5 + 2^3 = 5 + 8 = 13$$

Example 39: Evaluate $-3^2$.

This one catches a lot of people. The exponent applies only to the 3, not the negative sign: $$-3^2 = -(3^2) = -9$$

Think of the minus sign as subtraction happening after the exponent. If you wanted to square $-3$, you'd need parentheses: $(-3)^2 = 9$.

This is why $-x^2$ and $(-x)^2$ give different results. The first squares $x$ then makes it negative. The second squares the entire negative $x$.

One more thing: exponents don't distribute over addition or subtraction. So $(x + y)^2$ is NOT the same as $x^2 + y^2$. You'd have to multiply it out the long way: $(x + y)^2 = (x + y)(x + y) = x^2 + 2xy + y^2$. A lot of students fall into this trap, so watch out for it.

Why Exponents Matter

Exponents are everywhere in mathematics and science. They describe how things grow (population growth follows exponential patterns), how things shrink (radioactive decay), how areas and volumes scale (squaring and cubing dimensions), and how we express very large or very small numbers (scientific notation uses powers of 10).

Understanding exponents also prepares you for more advanced math. Polynomials are built from terms with exponents. Exponential and logarithmic functions (which you'll encounter in algebra and beyond) are based on these same principles. Even calculus relies heavily on manipulating expressions with exponents.

The rules you've learned here—the product rule, quotient rule, power rule, and how to handle zero and negative exponents—will serve you throughout your mathematical journey. Practice them until they become second nature, because you'll use them constantly from here on out.