Laws of Exponents

⚡ Laws of Exponents

What Is an Exponent?

An exponent tells you how many times to multiply a base by itself.

In the power $a^m$:

• $a$ is the base (the number being multiplied)
• $m$ is the exponent (how many times the base is used as a factor)

For example, $2^4 = 2 \times 2 \times 2 \times 2 = 16$. The base $2$ appears as a factor $4$ times.

Writing things out the long way works, but it gets messy fast — imagine writing $2^{20}$! The laws of exponents are shortcuts that let us multiply, divide, and stack powers without expanding everything.

🔑 Big Idea: Each law is just a fast way to count factors. If you ever forget a rule, you can rebuild it by writing out the multiplication.

✖️ The Product Rule

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

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

Why it works: $x^3 \times x^4 = (x \cdot x \cdot x)(x \cdot x \cdot x \cdot x) = x^7$. There are $3 + 4 = 7$ factors of $x$.

Example: $x^3 \times x^4 = x^{3+4} = x^7$

➗ The Quotient Rule

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

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

Why it works: dividing cancels matching factors from the top and bottom.

Example: $\dfrac{x^7}{x^3} = x^{7-3} = x^4$

⚠️ Same base required! These shortcuts only work when the bases match. You cannot combine $2^3 \times 5^4$ this way because $2 \neq 5$.

🔁 The Power Rule

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

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

Why it works: $(x^3)^4$ means $x^3$ used as a factor $4$ times: $x^3 \cdot x^3 \cdot x^3 \cdot x^3 = x^{12}$. That's $3 \times 4 = 12$ factors.

Example: $(x^3)^4 = x^{3 \times 4} = x^{12}$

🧠 Don't mix them up! Multiplying powers means add exponents, but raising a power to a power means multiply exponents.

✅ Concept Check

📝 Worked Examples

Let's apply the three main rules step by step.

Example 1 — Product Rule

Simplify $5^2 \times 5^6$.

• The base is the same ($5$), so add the exponents.
• $2 + 6 = 8$
• Answer: $5^8$

Example 2 — Quotient Rule

Simplify $\dfrac{a^9}{a^4}$.

• The base is the same ($a$), so subtract the exponents (top minus bottom).
• $9 - 4 = 5$
• Answer: $a^5$

Example 3 — Power Rule

Simplify $(b^4)^3$.

• A power raised to a power, so multiply the exponents.
• $4 \times 3 = 12$
• Answer: $b^{12}$

Example 4 — Two rules together

Simplify $\dfrac{x^2 \times x^5}{x^3}$.

• Step 1 — Product rule (top): $x^2 \times x^5 = x^{2+5} = x^7$
• Step 2 — Quotient rule: $\dfrac{x^7}{x^3} = x^{7-3} = x^4$
• Answer: $x^4$

✏️ Your Turn

Find the new exponent for each simplified power. Type just the number.

Box 1: $7^3 \times 7^5 = 7^{?}$ — what is the exponent? Box 2: $\dfrac{m^{10}}{m^6} = m^{?}$ — what is the exponent? Box 3: $(k^4)^2 = k^{?}$ — what is the exponent?

🎯 Guided Practice

Decide which rule applies, then add, subtract, or multiply the exponents.

🔽 Pick the Simplified Power

Choose the correct simplified form for each expression.

🌍 Two Special Exponents

Two more rules round out the laws of exponents. They show up constantly in scientific notation and science class.

Zero Exponent

Any non-zero number raised to the power $0$ equals $1$:

$a^0 = 1 \quad (a \neq 0)$

Why? Think of the quotient rule: $\dfrac{a^3}{a^3} = a^{3-3} = a^0$. But any number divided by itself is $1$, so $a^0 = 1$.

Example: $5^0 = 1$

Negative Exponent

A negative exponent means take the reciprocal (flip to the bottom):

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

Example: $2^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8}$

Real-World Connection: Bacteria 🦠

A single bacterium splits into $2$ every hour. After $h$ hours there are $2^h$ bacteria.

• After $3$ hours: $2^3 = 8$ bacteria.
• After $5$ hours: $2^5 = 32$ bacteria.
• The growth from hour $3$ to hour $5$ is $\dfrac{2^5}{2^3} = 2^{5-3} = 2^2 = 4$ times as many. Exponent rules make this easy!

✏️ Apply the Special Rules

Evaluate each expression. Type your answer as a number or a simple fraction (like 1/8).

Box 1: $9^0 = ?$ Box 2: $2^{-3} = ?$ (write it as a fraction) Box 3: A colony doubles each hour as $2^h$. How many bacteria after $4$ hours? Compute $2^4 = ?$

✅ Apply It

🏆 Review: All Five Laws

You now know every law of exponents! Here is the complete toolkit:

🔑 Quick check for which operation:

• Multiplying powers → add exponents
• Dividing powers → subtract exponents
• Power of a power → multiply exponents

When more than one rule applies, work from the inside out — simplify inside parentheses first, then combine.

🚀 Mixed Challenge