# Laws of exponents with integer exponents

> IB Mathematics AI SL · IB MATH AI SL
> Source: https://www.owlsprep.com/study/ib-math-ai-sl-u1-laws-of-exponents-with-rational/

This module covers the laws of exponents for integer powers: the product, quotient and power laws, the power of a product/quotient, and the zero and negative-integer exponent rules. These are foundational manipulation skills used throughout IB AI SL.

## Learning objectives

- Apply the product, quotient and power laws to expressions with integer exponents
- Simplify expressions using the zero-exponent and negative-integer-exponent rules
- Combine several exponent laws to simplify algebraic expressions
- Evaluate numerical powers with integer exponents

## The Laws of Integer Exponents

An exponent (or index) is a shorthand for repeated multiplication: $a^n$ means the base $a$ multiplied by itself $n$ times. When we combine powers of the **same base**, a small set of laws lets us simplify quickly without writing everything out.

- Product law: $a^m \times a^n = a^{m+n}$
- Quotient law: $\frac{a^m}{a^n} = a^{m-n}$ for $a \neq 0$
- Power of a power: $(a^m)^n = a^{mn}$
- Power of a product: $(ab)^n = a^n b^n$
- Power of a quotient: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ for $b \neq 0$

> **warning**
>
> The product and quotient laws only apply when the bases are the same. $a^2 \times b^3$ cannot be combined into a single power.

**Worked example:** Simplify $3x^4 \times 2x^5$ and $\frac{12y^7}{4y^2}$.

1. For $3x^4 \times 2x^5$, multiply the coefficients and add the exponents of $x$ (product law):
2. $$3x^4 \times 2x^5 = (3 \times 2)\,x^{4+5} = 6x^9$$
3. For $\frac{12y^7}{4y^2}$, divide the coefficients and subtract the exponents of $y$ (quotient law):
4. $$\frac{12y^7}{4y^2} = \frac{12}{4}\,y^{7-2} = 3y^5$$

*Calculator:* forbidden

## Zero and Negative Integer Exponents

**Zero and Negative Exponents** — For any $a \neq 0$ and positive integer $n$: $a^0 = 1$ and $a^{-n} = \frac{1}{a^n}$. A negative exponent means take the reciprocal, it does **not** make the value negative.

*Notation:* a^0, \; a^{-n}

*Example:* $5^0 = 1$ and $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$

These follow naturally from the quotient law. For example $\frac{a^3}{a^3} = a^{3-3} = a^0$, but any non-zero number divided by itself is $1$, so $a^0 = 1$. Similarly $\frac{a^2}{a^5} = a^{2-5} = a^{-3}$, which equals $\frac{1}{a^3}$.

**Worked example:** Write $4x^{-2}$ and $(3x)^{-1}$ using only positive exponents.

1. In $4x^{-2}$ the negative exponent applies only to $x$, not to the coefficient $4$:
2. $$4x^{-2} = \frac{4}{x^2}$$
3. In $(3x)^{-1}$ the whole bracket is raised to $-1$, so take the reciprocal of $3x$:
4. $$(3x)^{-1} = \frac{1}{3x}$$

*Calculator:* forbidden

## Combining the Laws

Most exam questions need two or three laws together. Work from the inside out: deal with brackets using the power laws first, then combine like bases with the product and quotient laws, and finally rewrite any negative exponents as positive.

**Worked example:** Simplify $(2x^3)^4 \times x^{-5}$, giving your answer with positive exponents.

1. Apply the power of a product rule to the bracket, raising both the coefficient and the power of $x$ to the 4th power:
2. $$(2x^3)^4 = 2^4 (x^3)^4 = 16x^{12}$$
3. Now multiply by $x^{-5}$ using the product law (add the exponents of $x$):
4. $$16x^{12} \times x^{-5} = 16x^{12+(-5)} = 16x^7$$

**Check your understanding**

Test your understanding:

1. What is the simplified value of $\frac{2^5}{2^2}$?

   - 2
   - 4
   - 8
   - 16

   *Why:* Correct! $\frac{2^5}{2^2} = 2^{5-2} = 2^3 = 8$. If you got 4, you may have divided the exponents instead of subtracting them.

*Calculator:* forbidden

## Common pitfalls

- **Wrong:** Combining different bases: $a^2 \times b^3 = (ab)^5$
  - Why it fails: The product law only adds exponents when the bases are identical
  - Correct: Leave unlike bases separate: $a^2 \times b^3$ is already fully simplified
- **Wrong:** Writing $a^0 = 0$
  - Why it fails: Any non-zero base raised to the power $0$ equals $1$, not $0$
  - Correct: State $a^0 = 1$ for $a \neq 0$
- **Wrong:** Treating a negative exponent as a negative value: $2^{-3} = -8$
  - Why it fails: A negative exponent means reciprocal, not a negative number
  - Correct: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$
- **Wrong:** Forgetting to raise the coefficient: $(3x)^2 = 3x^2$
  - Why it fails: The power of a product rule requires the exponent to apply to all factors, including coefficients
  - Correct: $(3x)^2 = 3^2 x^2 = 9x^2$

## Cheatsheet

| Rule Name | Integer Exponent Form | Example |
| --- | --- | --- |
| Product Law | $a^m a^n = a^{m+n}$ | $x^4 x^5 = x^9$ |
| Quotient Law | $\frac{a^m}{a^n} = a^{m-n}$ | $\frac{y^7}{y^2} = y^5$ |
| Power of a Power | $(a^m)^n = a^{mn}$ | $(z^3)^2 = z^6$ |
| Power of a Product | $(ab)^n = a^n b^n$ | $(3x)^2 = 9x^2$ |
| Zero Exponent | $a^0 = 1$ | $7^0 = 1$ |
| Negative Exponent | $a^{-n} = \frac{1}{a^n}$ | $2^{-3} = \frac{1}{8}$ |

## What's next

The laws of exponents are a foundational skill you will use across the entire IB AI SL syllabus. They underpin scientific notation, exponential functions used to model growth and decay, and geometric sequences and series. They also connect directly to logarithms, which reverse exponentiation. Mastering these laws now will make more advanced topics much easier to handle.

- [Operations with numbers in scientific notation](https://www.owlsprep.com/study/ib-math-ai-sl-u1-operations-with-numbers-in-scientific/)
- [Simplification of logarithmic expressions](https://www.owlsprep.com/study/ib-math-ai-sl-u1-simplification-of-logarithmic-expressions/)
- [Solving exponential equations using logarithms](https://www.owlsprep.com/study/ib-math-ai-sl-u1-solving-exponential-equations-using-logarithms/)

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