Laws of exponents

IB Mathematics: Applications and Interpretation HL· SL 1.5 / AHL 1.10 Laws of exponents· 18 min read

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

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An exponent (or index) tells you how many times a base is used as a factor. From this idea we derive a small set of laws that let us combine and simplify powers of the same base. These laws are on SL 1.5 and are assumed knowledge for all HL work.

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 $b \neq = 0$
Zero exponent: $a^{0} = 1$ for $a \neq = 0$
Negative exponent: $a^{- n} = \frac{1}{a ^{n}}$ for $a \neq = 0$

📐 Worked Example

Simplify $\frac{6 x ^{5} y ^{- 2}}{2 x ^{2} y ^{3}}$ , giving your answer with positive exponents only.

1
Separate the numerical part from each variable so you can apply the laws term by term:
2
$\frac{6}{2} \times \frac{x^5}{x^2} \times \frac{y^{-2}}{y^{3}}$
3
Divide the coefficients, and apply the quotient law to each variable (subtract exponents):
4
$3 \times x^{5-2} \times y^{-2-3} = 3x^{3} y^{-5}$
5
Rewrite the negative exponent as a reciprocal to give positive exponents only:
6
$3x^{3} y^{-5} = \frac{3x^{3}}{y^{5}}$

Exam tip:

Deal with coefficients and each variable separately, then recombine. This avoids the classic mistake of dividing coefficients using an exponent law.

2. Rational Exponents and Radicals★★★☆☆⏱ 6 min

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📘 Definition

Rational Exponent

a^{\frac{m}{n}}

For any non-negative real number $a$ , and integers $m, n$ with $n > 0$ : $a^{\frac{m}{n}} = n a^{m} = (n a)^{m}$ . The denominator $n$ is the root and the numerator $m$ is the power. If $n$ is even, $a$ must be non-negative for a real result.

Example:

$8^{\frac{2}{3}} = (38)^{2} = 2^{2} = 4$

AHL 1.10 extends the exponent laws to rational exponents. Every law from the integer case still holds; the only extra skill is doing fraction arithmetic when you add or subtract the exponents. A power of $\frac{1}{n}$ is exactly an nth root, which is what links this section to radical notation.

📐 Worked Example

Evaluate $1 6^{- \frac{3}{4}}$ without simplifying to a decimal.

1
Deal with the negative exponent first by taking the reciprocal:
2
$16^{-\frac{3}{4}} = \frac{1}{16^{\frac{3}{4}}}$
3
Interpret the rational exponent: denominator 4 is the fourth root, numerator 3 is the power:
4
$16^{\frac{3}{4}} = \left(\sqrt[4]{16}\right)^3 = 2^3 = 8$
5
Combine the two results:
6
$16^{-\frac{3}{4}} = \frac{1}{8}$

Exam tip:

When you see a negative fractional exponent, handle it in a fixed order: reciprocal first (the sign), then the root (denominator), then the power (numerator).

3. Simplifying Algebraic Expressions with Rational Exponents★★★☆☆⏱ 6 min

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In algebraic simplification a GDC gives you no help, so you must apply the laws confidently. Distribute an outer exponent to every factor inside the bracket, including the coefficient, then combine like bases.

📐 Worked Example

Simplify $(27 a^{6} b^{\frac{1}{2}})^{\frac{2}{3}}$ , writing your answer with positive exponents.

1
Apply the power of a product law: distribute the exponent $\frac{2}{3}$ to every factor:
2
$27^{\frac{2}{3}} \times \left(a^{6}\right)^{\frac{2}{3}} \times \left(b^{\frac{1}{2}}\right)^{\frac{2}{3}}$
3
Evaluate the numerical factor and use the power of a power law (multiply exponents) on each variable:
4
$27^{\frac{2}{3}} = \left(\sqrt[3]{27}\right)^2 = 3^2 = 9$
5
$\left(a^{6}\right)^{\frac{2}{3}} = a^{6 \times \frac{2}{3}} = a^{4}, \quad \left(b^{\frac{1}{2}}\right)^{\frac{2}{3}} = b^{\frac{1}{2} \times \frac{2}{3}} = b^{\frac{1}{3}}$
6
Combine all the simplified factors:
7
$\left(27a^{6} b^{\frac{1}{2}}\right)^{\frac{2}{3}} = 9a^{4} b^{\frac{1}{3}}$

✓ Quick check

Test your understanding:

What is the simplified value of $3 2^{\frac{2}{5}}$ ?
- 2
- 4
- 8
- 64
Reveal answer
4 —
Correct! $3 2^{\frac{2}{5}} = (532)^{2} = 2^{2} = 4$ . If you got 8, you took the fifth root then cubed instead of squaring.

Exam tip:

Check whether your final exponents should be positive: examiners often specify 'positive exponents only', so convert any negatives to reciprocals at the end.

4. Common Pitfalls

Wrong move:

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

Why:

A negative exponent produces a reciprocal, not a negative value.

Correct move:

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

Wrong move:

Adding rational exponents without a common denominator: $a^{\frac{1}{2}} a^{\frac{1}{3}} = a^{\frac{2}{5}}$

Why:

Fractions cannot be added by adding numerators and denominators separately.

Correct move:

$\frac{1}{2} + \frac{1}{3} = \frac{5}{6}$ , so $a^{\frac{1}{2}} a^{\frac{1}{3}} = a^{\frac{5}{6}}$

Wrong move:

Forgetting to raise the coefficient: $(9 x^{4})^{\frac{1}{2}} = 9 x^{2}$

Why:

The power of a product rule applies the outer exponent to every factor, including the number.

Correct move:

$(9 x^{4})^{\frac{1}{2}} = 9^{\frac{1}{2}} x^{2} = 3 x^{2}$

Wrong move:

Applying the wrong part of the fraction: reading $a^{\frac{2}{3}}$ as cube of the square root

Why:

The denominator is the root and the numerator is the power, so it is the square of the cube root.

Correct move:

$a^{\frac{2}{3}} = (3 a)^{2} = 3 a^{2}$

5. Quick Reference Cheatsheet

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

6. Frequently Asked

Do I need to memorise the exponent laws for IB AI HL?

Yes. The laws of exponents are not on the formula booklet, so you must know them from memory. A GDC can check a numerical value but cannot simplify an algebraic expression for you.

What is the difference between the SL and HL parts of this topic?

SL 1.5 covers the laws of exponents with integer exponents. AHL 1.10 extends this to rational (fractional) exponents and their link to radicals. HL students are examined on both.

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2023 · Paper 1
Simplify an expression with integer and negative exponents
2022 · Paper 1
Evaluate a numerical expression with a rational exponent

Going deeper

syllabusIB AI HL Official Syllabus (SL 1.5, AHL 1.10)Official subject reference

What's Next

The laws of exponents are a foundational skill you will reuse across the whole IB AI HL syllabus. They underpin exponential models of growth and decay, the manipulation of logarithms, standard form, and financial applications such as compound interest. A secure grasp of both integer and rational exponents now will make exponential and logarithmic topics much easier to handle later.

→Number and Algebra overview