Laws of exponents with integer exponents

IB Mathematics AI SL· SL 1.5· 12 min read

1. The Laws of Integer Exponents★★☆☆☆⏱ 5 min

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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: $(\frac{a}{b})^{n} = \frac{a ^{n}}{b ^{n}}$ for $b \neq = 0$

📐 Worked Example

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

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

2. Zero and Negative Integer Exponents★★☆☆☆⏱ 5 min

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📘 Definition

Zero and Negative Exponents

a^0, \; a^{-n}

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.

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 $4 x^{- 2}$ and $(3 x)^{- 1}$ using only positive exponents.

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

3. Combining the Laws★★★☆☆⏱ 6 min

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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 $(2 x^{3})^{4} \times x^{- 5}$ , 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
$16x^{12} \times x^{-5} = 16x^{12+(-5)} = 16x^7$

✓ Quick check

Test your understanding:

What is the simplified value of $\frac{2 ^{5}}{2 ^{2}}$ ?
- 2
- 4
- 8
- 16
Reveal answer
8 —
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.

4. Common Pitfalls

Wrong move:

Combining different bases: $a^{2} \times b^{3} = (ab)^{5}$

Why:

The product law only adds exponents when the bases are identical

Correct move:

Leave unlike bases separate: $a^{2} \times b^{3}$ is already fully simplified

Wrong move:

Writing $a^{0} = 0$

Why:

Any non-zero base raised to the power $0$ equals $1$ , not $0$

Correct move:

State $a^{0} = 1$ for $a \neq = 0$

Wrong move:

Treating a negative exponent as a negative value: $2^{- 3} = - 8$

Why:

A negative exponent means reciprocal, not a negative number

Correct move:

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

Wrong move:

Forgetting to raise the coefficient: $(3 x)^{2} = 3 x^{2}$

Why:

The power of a product rule requires the exponent to apply to all factors, including coefficients

Correct move:

$(3 x)^{2} = 3^{2} x^{2} = 9 x^{2}$

5. Quick Reference 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	$(ab)^{n} = a^{n} b^{n}$	$(3 x)^{2} = 9 x^{2}$
Zero Exponent	$a^{0} = 1$	$7^{0} = 1$
Negative Exponent	$a^{- n} = \frac{1}{a ^{n}}$	$2^{- 3} = \frac{1}{8}$

Going deeper

videoIB OSC: Laws of Exponents ReviewFree 10 minute revision summary

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.