Exponents (or powers) are a shorthand way to show how many times a number is multiplied by itself. Understanding these rules is the foundation for algebra, calculus, and scientific notation. Click on any rule card to see it highlighted!
1. Core Definitions
Before diving into the rules, let's define the terms:
$$b^n$$
$b$ (Base)
The number being multiplied
Example: In $5^3$, the base is $5$
$n$ (Exponent)
The number of times the base is used in multiplication
Example: In $5^3$, the exponent is $3$
💡 What does $5^3$ mean?
$$5^3 = 5 \times 5 \times 5 = 125$$
The base (5) is multiplied by itself 3 times.
2. Essential Exponent Rules
Click on each rule card to highlight it and see interactive examples!
✖️ Product Rule
$$a^m \cdot a^n = a^{m+n}$$
To multiply like bases, add the exponents.
Example:
$$x^2 \cdot x^3 = x^{2+3} = x^5$$
➗ Quotient Rule
$$\frac{a^m}{a^n} = a^{m-n}$$
To divide like bases, subtract the exponents.
Example:
$$\frac{y^9}{y^4} = y^{9-4} = y^5$$
🔺 Power of a Power
$$(a^m)^n = a^{m \cdot n}$$
To raise a power to a power, multiply exponents.
Example:
$$(2^3)^2 = 2^{3 \cdot 2} = 2^6 = 64$$
📦 Power of a Product
$$(ab)^n = a^n b^n$$
Distribute the exponent to every factor inside.
Example:
$$(xy)^3 = x^3 y^3$$
📊 Power of a Quotient
$$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$
Distribute the exponent to the top and bottom.
Example:
$$\left(\frac{x}{5}\right)^2 = \frac{x^2}{5^2} = \frac{x^2}{25}$$
🎮 Interactive Practice: Apply the Rules
Simplify: $x^4 \cdot x^2 \div x^3$
Step 1: Apply Product Rule to $x^4 \cdot x^2$
$$x^4 \cdot x^2 = x^{4+2} = x^6$$
Step 2: Apply Quotient Rule to $x^6 \div x^3$
$$\frac{x^6}{x^3} = x^{6-3} = x^3$$
Final Answer: $x^3$
3. Special Exponents (The "Tricky" Ones)
⓪ Zero Exponent Property
Any non-zero base raised to the power of zero is always 1.
$$a^0 = 1 \quad (a \neq 0)$$
Examples:
$$5^0 = 1$$
$$(1,245,678)^0 = 1$$
$$(xyz)^0 = 1$$
💡 Why? Using the quotient rule: $\frac{a^n}{a^n} = a^{n-n} = a^0$, but $\frac{a^n}{a^n} = 1$
➖ Negative Exponent Property
A negative exponent represents the reciprocal (the "flip") of the base.
$$a^{-n} = \frac{1}{a^n} \quad \text{and} \quad \frac{1}{a^{-n}} = a^n$$
Examples:
$$3^{-2} = \frac{1}{3^2} = \frac{1}{9}$$
$$\frac{1}{x^{-4}} = x^4$$
$$2^{-3} = \frac{1}{2^3} = \frac{1}{8}$$
🔢 Fractional (Rational) Exponents
Fractional exponents represent roots. The denominator is the index of the root, and the numerator is the power.
$$a^{\frac{m}{n}} = \sqrt[n]{a^m} \quad \text{or} \quad (\sqrt[n]{a})^m$$
Examples:
$$16^{3/4} = (\sqrt[4]{16})^3 = (2)^3 = 8$$
$$8^{2/3} = (\sqrt[3]{8})^2 = (2)^2 = 4$$
$$x^{1/2} = \sqrt{x}$$
🧪 Try It: Simplify $27^{2/3}$
Step 1: Recognize that $27 = 3^3$
Step 2: Apply the fractional exponent rule:
$$27^{2/3} = (\sqrt[3]{27})^2$$
Step 3: Simplify the cube root:
$$(\sqrt[3]{27})^2 = (3)^2 = 9$$
Answer: $9$
4. Quick Reference Table
Use this table as a quick cheat sheet for solving problems:
| If you see... | Do this to the exponents... |
|---|---|
| Multiplication ($x^a \cdot x^b$) | Add ($a + b$) |
| Division ($\frac{x^a}{x^b}$) | Subtract ($a - b$) |
| Parentheses ($(x^a)^b$) | Multiply ($a \cdot b$) |
| Negative ($x^{-a}$) | Flip to fraction ($\frac{1}{x^a}$) |
| Fraction ($x^{a/b}$) | Root ($\sqrt[b]{x^a}$) |
🎯 Quick Quiz
What operation do you perform on exponents when you see $(x^5)^3$?
Multiply the exponents!
$$(x^5)^3 = x^{5 \cdot 3} = x^{15}$$
5. Pro-Tips for Solving Problems
1️⃣ Order of Operations
Always follow PEMDAS. Handle exponents inside parentheses before applying an outside exponent.
Example: $(2 \cdot 3)^2 = 6^2 = 36$ ✓
Not: $2 \cdot 3^2 = 2 \cdot 9 = 18$ ✗
2️⃣ Keep the Base
When adding/subtracting exponents, the base never changes.
$$5^2 \cdot 5^3 = 5^5 = 3125$$ ✓
$$\text{Not: } 25^5$$ ✗
3️⃣ Simplify Negatives Last
It is usually easier to simplify the whole expression using product/quotient rules first, then flip any remaining negative exponents at the very end.
4️⃣ Coefficients vs. Exponents
Remember that coefficients multiply normally, but exponents add.
$$2x^3 \cdot 4x^2 = (2 \cdot 4)(x^{3+2}) = 8x^5$$ ✓
6. Practice Problems
Problem 1: Simplify $(3x^2y^3)^2$
Step 1: Apply Power of a Product rule
$$(3x^2y^3)^2 = 3^2 \cdot (x^2)^2 \cdot (y^3)^2$$
Step 2: Apply Power of a Power rule
$$= 9 \cdot x^{2 \cdot 2} \cdot y^{3 \cdot 2}$$
Step 3: Simplify
$$= 9x^4y^6$$
Problem 2: Simplify $\frac{a^8b^5}{a^3b^2}$
Step 1: Apply Quotient Rule to each variable
$$\frac{a^8}{a^3} \cdot \frac{b^5}{b^2}$$
Step 2: Subtract exponents
$$= a^{8-3} \cdot b^{5-2}$$
$$= a^5b^3$$
Problem 3: Simplify $\frac{x^{-3}y^4}{x^2y^{-1}}$
Step 1: Apply Quotient Rule
$$x^{-3-2} \cdot y^{4-(-1)}$$
Step 2: Simplify exponents
$$= x^{-5} \cdot y^{5}$$
Step 3: Apply Negative Exponent Rule
$$= \frac{y^5}{x^5}$$
Problem 4: Simplify $64^{2/3}$
Step 1: Recognize that $64 = 4^3$
Step 2: Apply Fractional Exponent Rule
$$64^{2/3} = (\sqrt[3]{64})^2$$
Step 3: Simplify the cube root
$$= (4)^2$$
$$= 16$$
🌟 Challenge Problem
Simplify: $\frac{(2x^3y^{-2})^3}{4x^5y^{-4}}$
Step 1: Apply Power of a Product to numerator
$$\frac{2^3 \cdot (x^3)^3 \cdot (y^{-2})^3}{4x^5y^{-4}}$$
Step 2: Apply Power of a Power
$$= \frac{8 \cdot x^9 \cdot y^{-6}}{4x^5y^{-4}}$$
Step 3: Simplify coefficient and apply Quotient Rule
$$= 2 \cdot x^{9-5} \cdot y^{-6-(-4)}$$
Step 4: Simplify exponents
$$= 2x^4y^{-2}$$
Step 5: Apply Negative Exponent Rule
$$= \frac{2x^4}{y^2}$$