Exponents and Exponential Expressions

You have almost certainly seen exponents before. That little number floating above and to the right of another number is just a shorthand you probably already understand intuitively. When you write $5 \times 5 \times 5$ over and over again, it gets tedious. Exponents solve that problem by letting you write $5^3$ instead. That is it. No mystery, no trick. Exponents are just a way to save time and space when you need to multiply a number by itself repeatedly.

In Algebra 1, we take this familiar idea and build a complete toolkit around it. You will discover patterns that let you simplify complicated-looking expressions in just a few steps. The good news is that these patterns are not arbitrary rules to memorize; they are logical consequences of what exponents actually mean. Once you see why they work, you will not need to memorize them at all.

Core Concepts

Exponent Review

An exponent tells you how many times to multiply a number by itself. In the expression $a^n$:

$$a^n = \underbrace{a \times a \times a \times \cdots \times a}_{n \text{ times}}$$

So when you see $x^4$, you know it means $x \times x \times x \times x$. The number being multiplied (in this case, $x$) is called the base, and the number telling you how many times to multiply (in this case, 4) is called the exponent.

The Product Rule

What happens when you multiply two powers that have the same base? Let us look at an example:

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

Do you see the pattern? We had 3 copies of $x$ times 2 more copies of $x$, giving us 5 copies total. This is the product rule:

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

When you multiply powers with the same base, add the exponents.

The Quotient Rule

Now suppose you divide powers with the same base:

$$\frac{x^5}{x^2} = \frac{x \times x \times x \times x \times x}{x \times x} = x \times x \times x = x^3$$

Two of the $x$’s in the numerator cancel with the two in the denominator, leaving us with three. This is the quotient rule:

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

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

The Power Rule

What if you raise a power to another power? Consider $(x^2)^3$:

$$(x^2)^3 = x^2 \times x^2 \times x^2 = x^{2+2+2} = x^6$$

We are multiplying $x^2$ by itself 3 times, so we add the exponent 2 three times, which is the same as multiplying $2 \times 3$. This is the power rule:

$$(a^m)^n = a^{mn}$$

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

Zero Exponents

What does $a^0$ mean? Using the quotient rule:

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

But we also know that any number divided by itself equals 1. Therefore:

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

Any nonzero number raised to the power of zero equals 1. This is not an arbitrary definition; it follows directly from the quotient rule.

Negative Exponents

What about $a^{-3}$? Again, let us use the quotient rule:

$$\frac{a^2}{a^5} = a^{2-5} = a^{-3}$$

But we can also simplify this fraction directly:

$$\frac{a^2}{a^5} = \frac{a \times a}{a \times a \times a \times a \times a} = \frac{1}{a^3}$$

So we discover that:

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

A negative exponent means “take the reciprocal.” The negative sign does not make the result negative; it flips the number to the other side of a fraction bar.

Products and Quotients to Powers

When you raise a product to a power, the exponent applies to each factor:

$$(ab)^n = a^n b^n$$

Why? $(ab)^3 = (ab)(ab)(ab) = (a \cdot a \cdot a)(b \cdot b \cdot b) = a^3 b^3$

Similarly, when you raise a quotient to a power:

$$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$

Scientific Notation

When numbers get extremely large or extremely small, writing them out becomes impractical. Scientific notation uses exponents to express these numbers compactly:

$$a \times 10^n$$

where $1 \leq a < 10$ and $n$ is an integer.

For large numbers, the exponent is positive. Each time you move the decimal point one place to the left, you increase the exponent by 1:

$$45,000,000 = 4.5 \times 10^7$$

For small numbers, the exponent is negative. Each time you move the decimal point one place to the right, you decrease the exponent by 1:

$$0.000032 = 3.2 \times 10^{-5}$$

Notation and Terminology

Term	Meaning	Example
Base	Number being multiplied	In $5^3$, base is 5
Exponent	Number of times base is multiplied	In $5^3$, exponent is 3
Power	The result of exponentiation	$5^3 = 125$
Scientific notation	$a \times 10^n$ where $1 \leq a < 10$	$3.2 \times 10^5 = 320000$

Examples

Key Properties and Rules

Here is a summary of all the exponent rules in one place:

Rule	Formula	Example
Product Rule	$a^m \cdot a^n = a^{m+n}$	$x^2 \cdot x^5 = x^7$
Quotient Rule	$\frac{a^m}{a^n} = a^{m-n}$	$\frac{x^7}{x^3} = x^4$
Power Rule	$(a^m)^n = a^{mn}$	$(x^3)^2 = x^6$
Zero Exponent	$a^0 = 1$	$5^0 = 1$
Negative Exponent	$a^{-n} = \frac{1}{a^n}$	$x^{-2} = \frac{1}{x^2}$
Product to a Power	$(ab)^n = a^n b^n$	$(2x)^3 = 8x^3$
Quotient to a Power	$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$	$\left(\frac{x}{3}\right)^2 = \frac{x^2}{9}$

Important note: These rules only work when the bases are the same. You cannot use the product rule to simplify $x^3 \cdot y^2$ because the bases are different.

Real-World Applications

Population Growth

If a population doubles every year, you can model the growth with exponents. Starting with 1000 individuals:

$$\text{Population after } n \text{ years} = 1000 \times 2^n$$

After 10 years: $1000 \times 2^{10} = 1000 \times 1024 = 1,024,000$

Compound Interest

When money earns compound interest, exponents determine how quickly it grows. If you invest $5000 at 6% annual interest:

$$\text{Value after } n \text{ years} = 5000 \times (1.06)^n$$

After 20 years: $\$5000 \times (1.06)^{20} \approx \$16,036$

Computer Memory

Computer storage is built on powers of 2:

• 1 kilobyte (KB) = $2^{10}$ = 1,024 bytes
• 1 megabyte (MB) = $2^{20}$ = 1,048,576 bytes
• 1 gigabyte (GB) = $2^{30}$ = 1,073,741,824 bytes
• 1 terabyte (TB) = $2^{40}$ = about 1 trillion bytes

This is why your “500 GB” hard drive does not quite hold 500 billion bytes.

Scientific Measurements

Scientists use scientific notation constantly because they work with numbers that are either extremely large or extremely small:

• Distance to the nearest star: $4 \times 10^{13}$ kilometers
• Mass of an electron: $9.1 \times 10^{-31}$ kilograms
• Avogadro’s number (atoms in a mole): $6.02 \times 10^{23}$

Without scientific notation, these numbers would be nearly impossible to read, write, or work with.

Self-Test Problems

Summary

• Exponents represent repeated multiplication: $a^n$ means $a$ multiplied by itself $n$ times
• Product rule: When multiplying powers with the same base, add exponents: $a^m \cdot a^n = a^{m+n}$
• Quotient rule: When dividing powers with the same base, subtract exponents: $\frac{a^m}{a^n} = a^{m-n}$
• Power rule: When raising a power to a power, multiply exponents: $(a^m)^n = a^{mn}$
• Zero exponent: Any nonzero number raised to the zero power equals 1: $a^0 = 1$
• Negative exponent: A negative exponent means take the reciprocal: $a^{-n} = \frac{1}{a^n}$
• Products and quotients to powers: Distribute the exponent to each factor: $(ab)^n = a^n b^n$
• Scientific notation: Expresses numbers as $a \times 10^n$ where $1 \leq a < 10$
• All exponent rules follow logically from the basic definition; understanding why they work is more valuable than memorizing them