Exponents and Roots

If you have ever felt a little intimidated when you see numbers with tiny numbers floating above them, or that strange checkmark-looking symbol with numbers inside, you are definitely not alone. Exponents and roots can look like some kind of secret mathematical code. But here is the good news: you already understand the core idea behind exponents. Every time you have calculated the area of a square or figured out how many items are in a grid, you have been using the concept of exponents without even knowing it.

In this chapter, we are going to demystify these symbols and show you that exponents and roots are simply shortcuts for operations you already know how to do. By the end, you will be reading and writing these expressions with confidence.

Core Concepts

What Are Exponents?

An exponent is a shorthand way of writing repeated multiplication. Just like multiplication is a shortcut for repeated addition (5 + 5 + 5 = 5 x 3), an exponent is a shortcut for repeated multiplication.

When you see:

$$2^4$$

This means “multiply 2 by itself 4 times”:

$$2^4 = 2 \times 2 \times 2 \times 2 = 16$$

That is it. No magic, no secrets. Exponents are just a compact way to write out what would otherwise be a long multiplication problem.

What Are Roots?

Roots are the opposite of exponents. If exponents are about repeated multiplication, roots ask the question: “What number, when multiplied by itself a certain number of times, gives me this result?”

For example, the square root of 16 asks: “What number times itself equals 16?”

$$\sqrt{16} = 4 \quad \text{because} \quad 4 \times 4 = 16$$

Think of exponents and roots as inverse operations, just like addition and subtraction or multiplication and division. They undo each other.

Notation and Terminology

Reading Exponents

In the expression $a^n$:

• a is called the base (the number being multiplied)
• n is called the exponent or power (how many times to multiply)

Here is how to read common expressions:

Expression	How to Read It	What It Means
$3^2$	“three squared” or “three to the second power”	$3 \times 3$
$5^3$	“five cubed” or “five to the third power”	$5 \times 5 \times 5$
$2^4$	“two to the fourth power”	$2 \times 2 \times 2 \times 2$
$10^6$	“ten to the sixth power”	$10 \times 10 \times 10 \times 10 \times 10 \times 10$

We say “squared” for the exponent 2 because it relates to the area of a square. We say “cubed” for the exponent 3 because it relates to the volume of a cube. For any other exponent, we just say “to the ___ power.”

Reading Roots

Expression	How to Read It	What It Means
$\sqrt{9}$	“the square root of 9”	What number times itself equals 9?
$\sqrt[3]{8}$	“the cube root of 8”	What number times itself 3 times equals 8?
$\sqrt[4]{16}$	“the fourth root of 16”	What number times itself 4 times equals 16?

The small number in the “crook” of the root symbol is called the index. For square roots, we usually do not write the 2 because it is understood.

Evaluating Powers

Let us work through some examples of calculating powers step by step.

Key Properties and Rules

Here is where exponents become really powerful. Instead of always writing out the multiplication, we can use these rules to simplify expressions quickly.

The Product Rule (Multiplying Powers with the Same Base)

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

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

Why does this work? Think about it: $$2^3 \times 2^4 = (2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2) = 2^7$$

You have 3 twos times 4 twos, which is 7 twos total.

The Quotient Rule (Dividing Powers with the Same Base)

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

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

Why? If you have $\frac{2^5}{2^3}$, you can cancel out common factors: $$\frac{2 \times 2 \times 2 \times 2 \times 2}{2 \times 2 \times 2} = 2 \times 2 = 2^2$$

The Power Rule (Raising a Power to a Power)

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

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

Why? $(2^3)^2$ means $2^3 \times 2^3 = 2^{3+3} = 2^6$.

Zero and Negative Exponents

What happens when the exponent is zero or negative?

Zero exponent: Any non-zero number raised to the power of zero equals 1.

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

Why? Using the quotient rule: $\frac{a^n}{a^n} = a^{n-n} = a^0$. But we also know that any number divided by itself is 1. So $a^0 = 1$.

Negative exponents: A negative exponent means “take the reciprocal.”

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

For example: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$

Why? Using the quotient rule: $\frac{2^2}{2^5} = 2^{2-5} = 2^{-3}$. But also: $\frac{2^2}{2^5} = \frac{4}{32} = \frac{1}{8}$.

Square Roots and Cube Roots

Perfect Squares

A perfect square is a number that results from squaring a whole number. Memorizing these will make your math life much easier:

$n$	$n^2$
1	1
2	4
3	9
4	16
5	25
6	36
7	49
8	64
9	81
10	100
11	121
12	144

When you need the square root of a perfect square, you can just work backwards from this table: $$\sqrt{49} = 7 \quad \text{because} \quad 7^2 = 49$$

Perfect Cubes

Similarly, perfect cubes result from cubing a whole number:

$n$	$n^3$
1	1
2	8
3	27
4	64
5	125
6	216

So $\sqrt[3]{27} = 3$ because $3^3 = 27$.

Estimating Roots

What if you need the square root of a number that is not a perfect square, like $\sqrt{50}$?

Strategy: Find the two perfect squares that your number falls between.

• $49 = 7^2$
• $64 = 8^2$

Since 50 is between 49 and 64, we know $\sqrt{50}$ is between 7 and 8. Since 50 is very close to 49, the answer is just a little more than 7. (The actual value is about 7.07.)

Scientific Notation

When numbers get very large or very small, writing them out becomes impractical. Imagine writing out the distance to the nearest star (about 40,000,000,000,000 kilometers) or the mass of a hydrogen atom (about 0.00000000000000000000000000167 kilograms). Scientific notation uses exponents to express these numbers in a compact form.

The Format

A number in scientific notation looks like:

$$a \times 10^n$$

Where:

• $a$ is a number between 1 and 10 (including 1, but not 10)
• $n$ is an integer (positive, negative, or zero)

Converting to Scientific Notation

For large numbers: Move the decimal point left until you have a number between 1 and 10. Count how many places you moved; that becomes your positive exponent.

$$5,280,000 = 5.28 \times 10^6$$

(We moved the decimal 6 places to the left.)

For small numbers: Move the decimal point right until you have a number between 1 and 10. Count how many places you moved; that becomes your negative exponent.

$$0.00047 = 4.7 \times 10^{-4}$$

(We moved the decimal 4 places to the right.)

Converting from Scientific Notation

Positive exponent: Move the decimal point to the right. $$3.2 \times 10^5 = 320,000$$

Negative exponent: Move the decimal point to the left. $$6.1 \times 10^{-3} = 0.0061$$

Operations with Scientific Notation

Multiplying: Multiply the coefficients and add the exponents. $$(3 \times 10^4) \times (2 \times 10^5) = 6 \times 10^9$$

Dividing: Divide the coefficients and subtract the exponents. $$\frac{8 \times 10^7}{2 \times 10^3} = 4 \times 10^4$$

Real-World Applications

Money and Compound Interest

When your money earns compound interest, exponents determine how it grows. If you invest $1000 at 5% annual interest, after $n$ years you have:

$$\$1000 \times (1.05)^n$$

After 10 years: $\$1000 \times (1.05)^{10} \approx \$1,628.89$

This is why financial advisors say to start saving early; the exponent (time) has a dramatic effect.

Computer Storage

Computer memory is measured in powers of 2:

• 1 kilobyte (KB) = $2^{10}$ = 1,024 bytes
• 1 megabyte (MB) = $2^{20}$ = 1,048,576 bytes
• 1 gigabyte (GB) = $2^{30}$ = about 1 billion bytes

Area and Volume

• The area of a square with side length $s$ is $s^2$ (hence “squared”)
• The volume of a cube with side length $s$ is $s^3$ (hence “cubed”)

Earthquakes

The Richter scale uses logarithms (which are related to exponents). An earthquake that measures 6 on the Richter scale releases about 31.6 times more energy than one that measures 5. Each whole number increase represents a tenfold increase in wave amplitude.

Bacteria Growth

Bacteria often double at regular intervals. If you start with 100 bacteria that double every hour, after $h$ hours you have:

$$100 \times 2^h \text{ bacteria}$$

After 10 hours: $100 \times 2^{10} = 100 \times 1024 = 102,400$ bacteria

Self-Test Problems

Summary

• Exponents are shorthand for repeated multiplication. In $a^n$, $a$ is the base and $n$ is the exponent.
• Product rule: $a^m \times a^n = a^{m+n}$ (add exponents when multiplying)
• Quotient rule: $\frac{a^m}{a^n} = a^{m-n}$ (subtract exponents when dividing)
• Power rule: $(a^m)^n = a^{m \times n}$ (multiply exponents when raising a power to a power)
• Zero exponent: $a^0 = 1$ for any $a \neq 0$
• Negative exponent: $a^{-n} = \frac{1}{a^n}$
• Square roots and cube roots are the inverse of squaring and cubing
• Memorizing perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144) and perfect cubes (1, 8, 27, 64, 125) makes root calculations faster
• Scientific notation expresses numbers as $a \times 10^n$ where $1 \leq a < 10$
• Exponents appear everywhere: finance, science, computing, and measurement