Integer Exponents

⚡ Integer Exponents

You already know how to work with positive exponents like $2^3$ or $5^4$. But exponents are even more powerful than that! When we allow the exponent to be any integer — positive, zero, or negative — we unlock a whole new set of tools.

What Are Integer Exponents?

Integer exponents simply means the exponent can be any whole number, including $0$ and the negatives:

• Positive integers: $2^3$, $5^4$, $10^2$
• Zero: $5^0$, $10^0$, $x^0$
• Negative integers: $2^{-3}$, $5^{-2}$, $10^{-1}$

The amazing part is that all of them follow the same consistent rules, so once you learn the pattern, you can handle any of them.

Quick Review: Positive Exponents

In the power $a^n$:

• $a$ is the base — the number being multiplied
• $n$ is the exponent — how many times to use the base as a factor

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

🔑 Big Idea: An exponent counts repeated multiplication. Zero and negative exponents extend that idea in a way that keeps every rule working perfectly.

0️⃣ Zero as an Exponent

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

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

Examples:

• $5^0 = 1$
• $10^0 = 1$
• $237^0 = 1$
• $(-8)^0 = 1$
• $x^0 = 1$

Why is this true? Use the quotient rule for dividing powers:

• $\dfrac{5^3}{5^3} = \dfrac{125}{125} = 1$ — any number divided by itself is $1$.
• But the quotient rule also says $\dfrac{5^3}{5^3} = 5^{3-3} = 5^0$.
• Since both results must be equal, $5^0 = 1$.

⚠️ Special note: $0^0$ is undefined in mathematics — the zero-exponent rule only works for bases that are not zero.

➖ Negative Exponents

Rule: A negative exponent means take the reciprocal — flip the base to the bottom of a fraction and make the exponent positive.

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

Examples:

🧠 Watch out: A negative exponent does not make the answer negative! $2^{-3} = \dfrac{1}{8}$, which is a small positive number. The negative sign tells you to flip, not to subtract.

✅ Concept Check

📝 Worked Examples

Let's evaluate several integer-exponent expressions step by step.

Example 1 — Zero exponent

Evaluate $9^0$.

• Any non-zero base to the power $0$ is $1$.
• Answer: $9^0 = 1$

Example 2 — Negative exponent

Evaluate $4^{-2}$.

• Step 1 — Flip: $4^{-2} = \dfrac{1}{4^2}$
• Step 2 — Square the base: $4^2 = 16$
• Answer: $\dfrac{1}{16}$

Example 3 — Negative exponent with a fraction base

Evaluate $\left(\dfrac{2}{3}\right)^{-2}$.

• Rule: $\left(\dfrac{a}{b}\right)^{-n} = \left(\dfrac{b}{a}\right)^{n}$ — flip the fraction, then make the exponent positive.
• Step 1 — Flip the fraction: $\left(\dfrac{3}{2}\right)^{2}$
• Step 2 — Apply the exponent: $\dfrac{3^2}{2^2} = \dfrac{9}{4}$
• Answer: $\dfrac{9}{4}$

Example 4 — Negative exponent in a denominator

Evaluate $\dfrac{1}{2^{-3}}$.

• Rule: $\dfrac{1}{a^{-n}} = a^{n}$ — a negative exponent in the denominator jumps to the top and turns positive.
• $\dfrac{1}{2^{-3}} = 2^3 = 8$
• Answer: $8$

✏️ Your Turn

Evaluate each expression. Type your answer as a whole number or a simple fraction (like 1/16).

Box 1: $7^0 = ?$ Box 2: $5^{-2} = ?$ Box 3: $\left(\dfrac{1}{4}\right)^{-1} = ?$

🎯 Guided Practice

For each one, decide whether the exponent is zero or negative, then evaluate.

🔽 Rewrite Without Negative Exponents

Choose the equivalent form for each expression.

🌍 Where Integer Exponents Show Up

Negative and zero exponents aren't just abstract rules — they describe the real world, especially when measuring very small things.

Tiny measurements 🔬

Scientists describe small lengths using negative powers of $10$:

• $10^{-1} = \dfrac{1}{10} = 0.1$ (one tenth)
• $10^{-2} = \dfrac{1}{100} = 0.01$ (one hundredth)
• $10^{-3} = \dfrac{1}{1000} = 0.001$ (one thousandth)

A grain of sand is about $10^{-3}$ meters wide, and a virus is about $10^{-7}$ meters wide. Negative exponents let us write these without a long string of zeros.

Combining the laws of exponents

All the exponent rules still work with integer exponents:

🔑 Strategy: First combine exponents using the laws, then handle any negative result by flipping to a fraction at the very end.

✏️ Apply the Rules

Use the laws of exponents, then write each final answer as a whole number or simple fraction (like 1/4).

Box 1: $2^3 \cdot 2^{-5}$ — first the exponent becomes $2^{-2}$. What is the final value? Box 2: $\dfrac{6^2}{6^4}$ — simplify and evaluate. Box 3: A bacterium is about $10^{-6}$ meters wide. Written as a fraction $\dfrac{1}{?}$, what number goes in the blank?

🤔 Apply It

🏆 Review & Summary

You can now work with every integer exponent — positive, zero, and negative. Here is the whole toolkit in one place.

🔑 Remember the two big ideas:

1. A zero exponent always gives $1$ (for a non-zero base).
2. A negative exponent means flip, not subtract — the answer stays positive.

Now put it all together in the challenge below!

🎓 Mixed Challenge