Rules for Calculating with Powers: Complete Overview of Formulas

Mathematics in 9th grade requires mastering the fundamental formulas for powers. Once you understand them, simplifying expressions will be a breeze. In this article, you'll find an overview of all rules with explanations and examples.

What is a power
Rules of powers – overview
1. Multiplying powers with the same base
2. Dividing powers with the same base
3. Raising a power to a power
4. Multiplying powers with different bases
5. Raising a product to a power
Common mistakes when calculating with powers
Quick reference of all rules
Ready for the test?
Frequently Asked Questions (FAQ)

What is a power

A power is a shorthand notation for repeated multiplication of the same number.

a^{n} = n times a \cdot a \cdot \dots \cdot a

In this notation:

$a$ is the base of the power – the number being multiplied
$n$ is the exponent – how many times we multiply

Example: $2^{4} = 2 \cdot 2 \cdot 2 \cdot 2 = 16$

> 💡 The base is what we multiply, the exponent tells us how many times.

Rules of powers overview

When working with powers, several important rules apply. These rules will make your calculations easier and help simplify expressions:

Multiplication: $a^{m} \cdot a^{n} = a^{m + n}$ (same base)
Division: $a^{m} : a^{n} = a^{m - n}$ (same base)
Raising to a power: $(a^{m})^{n} = a^{m \cdot n}$ (always)
Product raised to a power: $(a \cdot b)^{n} = a^{n} \cdot b^{n}$ (always)

> 👉 For a basic introduction to powers, see: Powers – complete guide

1. Multiplying powers with the same base

When multiplying powers with the same base, keep the base and add the exponents.

a^{n} \cdot a^{m} = a^{n + m}

Why does this work?

a^{3} \cdot a^{2} = (a \cdot a \cdot a) \cdot (a \cdot a) = a \cdot a \cdot a \cdot a \cdot a = a^{5}

Examples

2^{3} \cdot 2^{2} = 2^{3 + 2} = 2^{5} = 32

5^{4} \cdot 5^{1} = 5^{4 + 1} = 5^{5} = 3125

1 0^{2} \cdot 1 0^{3} = 1 0^{2 + 3} = 1 0^{5} = 100000

Tip: Only look at the exponents – the base doesn't change!

Exercise

👉 Examples on multiplying powers

2. Dividing powers with the same base

When dividing powers with the same base, keep the base and subtract the exponents.

a^{n} : a^{m} = a^{n - m}

Examples

2^{5} : 2^{3} = 2^{5 - 3} = 2^{2} = 4

1 0^{6} : 1 0^{2} = 1 0^{6 - 2} = 1 0^{4} = 10000

Watch out for negative exponents!

The resulting exponent can also be negative. In that case, you get a fraction:

2^{2} : 2^{5} = 2^{2 - 5} = 2^{- 3} = \frac{1}{2 ^{3}}

> 👉 What does a negative exponent mean? Negative exponents – explanation

3. Raising a power to a power

When raising an existing power to another power, multiply the exponents.

(a^{n})^{m} = a^{n \cdot m}

Why does this work?

(a^{2})^{3} = a^{2} \cdot a^{2} \cdot a^{2} = a^{2 + 2 + 2} = a^{6} = a^{2 \cdot 3}

Examples

(2^{3})^{2} = 2^{3 \cdot 2} = 2^{6} = 64

(5^{2})^{3} = 5^{2 \cdot 3} = 5^{6} = 15625

(1 0^{2})^{2} = 1 0^{2 \cdot 2} = 1 0^{4} = 10000

> ⚠️ Common mistake: $(a^{2})^{3} \neq = a^{5}$ ! The exponents are multiplied ( $2 \cdot 3 = 6$ ), not added!

4. Multiplying powers with different bases

What if the powers have different bases? In this case, you must first calculate each power individually:

2^{3} \cdot 3^{2} = 8 \cdot 9 = 72

But watch out! If the exponent is the same, you can use this rule:

a^{n} \cdot b^{n} = (a \cdot b)^{n}

Example: $2^{2} \cdot 5^{2} = (2 \cdot 5)^{2} = 1 0^{2} = 100$

5. Raising a product to a power

When raising a product (expression in parentheses) to a power, we raise each factor to that power:

(a \cdot b)^{n} = a^{n} \cdot b^{n}

Examples

(2 \cdot 3)^{2} = 2^{2} \cdot 3^{2} = 4 \cdot 9 = 36

(2 \cdot 5)^{3} = 2^{3} \cdot 5^{3} = 8 \cdot 125 = 1000

Common mistakes when calculating with powers

Mistake 1: Adding exponents with different bases

a^{3} \cdot b^{2} \neq = a^{5} \cdot b^{5}

You can only add exponents if the base is the same!

Mistake 2: Multiplying instead of adding exponents

a^{3} \cdot a^{2} \neq = a^{6} (correct: a^{3 + 2} = a^{5})

Mistake 3: Adding exponents when raising to a power

(a^{3})^{2} \neq = a^{5} (correct: a^{3 \cdot 2} = a^{6})

Mistake 4: Confusing the signs

Remember:

$(- 2)^{2} = 4$ (positive, because we square -2)
$- 2^{2} = - 4$ (negative, because we square 2 and then apply the minus)

Quick reference of all rules

Quick reference:

Multiplication (same base): $a^{m} \cdot a^{n} = a^{m + n}$ → Example: $2^{3} \cdot 2^{2} = 2^{5} = 32$
Division (same base): $a^{m} : a^{n} = a^{m - n}$ → Example: $2^{5} : 2^{3} = 2^{2} = 4$
Power of a power: $(a^{m})^{n} = a^{m \cdot n}$ → Example: $(2^{3})^{2} = 2^{6} = 64$
Product to a power: $(a \cdot b)^{n} = a^{n} \cdot b^{n}$ → Example: $(2 \cdot 3)^{2} = 4 \cdot 9 = 36$
Same exponent, different bases: $a^{n} \cdot b^{n} = (a \cdot b)^{n}$ → Example: $2^{2} \cdot 5^{2} = (2 \cdot 5)^{2} = 100$

Ready for the test

Test yourself with our interactive exercises:

Interactive exercises

Frequently Asked Questions (FAQ)

Can I add $a^{3} + a^{2}$ ?

No. Adding powers is only possible if they are exactly the same term. $a^{3} + a^{2}$ cannot be simplified because the exponents are different.

What is the difference between $a^{m} \cdot a^{n}$ and $(a^{m})^{n}$ ?

$a^{m} \cdot a^{n}$ – we're multiplying two different powers with the same base. We add the exponents: $a^{m + n}$
$(a^{m})^{n}$ – we're raising a power to another power. We multiply the exponents: $a^{m \cdot n}$

Why does $a^{0} = 1$ ?

From the division rule: $a^{3} : a^{3} = a^{3 - 3} = a^{0}$ . But we also know that any number divided by itself equals 1. Therefore $a^{0} = 1$ .

👉 More about zero exponent

When do I get a negative exponent?

When dividing powers where the divisor's exponent is larger: $a^{2} : a^{5} = a^{2 - 5} = a^{- 3} = \frac{1}{a ^{3}}$

👉 More about negative exponents

Rules for Calculating with Powers: All Formulas and Examples

Rules for Calculating with Powers: Complete Overview of Formulas

Table of Contents

What is a power

Rules of powers overview

1. Multiplying powers with the same base

Why does this work?

Examples

Exercise

2. Dividing powers with the same base

Examples

Watch out for negative exponents!

3. Raising a power to a power

Why does this work?

Examples

4. Multiplying powers with different bases

5. Raising a product to a power

Examples

Common mistakes when calculating with powers

Mistake 1: Adding exponents with different bases

Mistake 2: Multiplying instead of adding exponents

Mistake 3: Adding exponents when raising to a power

Mistake 4: Confusing the signs

Quick reference of all rules

Ready for the test

Interactive exercises

Frequently Asked Questions (FAQ)

Can I add $a^{3} + a^{2}$ ?

What is the difference between $a^{m} \cdot a^{n}$ and $(a^{m})^{n}$ ?

Why does $a^{0} = 1$ ?

When do I get a negative exponent?

Rules for Calculating with Powers: All Formulas and Examples

Rules for Calculating with Powers: Complete Overview of Formulas

Table of Contents

What is a power

Rules of powers overview

1. Multiplying powers with the same base

Why does this work?

Examples

Exercise

2. Dividing powers with the same base

Examples

Watch out for negative exponents!

3. Raising a power to a power

Why does this work?

Examples

4. Multiplying powers with different bases

5. Raising a product to a power

Examples

Common mistakes when calculating with powers

Mistake 1: Adding exponents with different bases

Mistake 2: Multiplying instead of adding exponents

Mistake 3: Adding exponents when raising to a power

Mistake 4: Confusing the signs

Quick reference of all rules

Ready for the test

Interactive exercises

Frequently Asked Questions (FAQ)

Can I add a3+a2?

What is the difference between am⋅an and (am)n?

Why does a0=1?

When do I get a negative exponent?

Can I add $a^{3} + a^{2}$ ?

What is the difference between $a^{m} \cdot a^{n}$ and $(a^{m})^{n}$ ?

Why does $a^{0} = 1$ ?