What are the exponent rules?

With the same base: multiply powers by adding the exponents, divide them by subtracting the exponents, and raise a power to a power by multiplying the exponents. Any non-zero number to the power 0 is 1, and a negative exponent means one over the positive power.

What is the difference between 3² and 3 × 2?

3² means 3 × 3, which is 9. 3 × 2 is 6. The exponent tells you how many times the base is used as a factor.

Why is any number to the power 0 equal to 1?

Dividing a power by itself gives 1, and the dividing rule gives an exponent of 0. For example, 4³ ÷ 4³ = 64 ÷ 64 = 1, and the rule says 4³ ÷ 4³ = 4⁰, so 4⁰ = 1.

What are square and cube numbers?

A square number is a whole number multiplied by itself, such as 6² = 36. A cube number is a whole number used as a factor three times, such as 6³ = 216.

What does a negative exponent mean?

One over the positive power. 10⁻² = 1/10² = 1/100.

What grade learns exponents?

In the US, grade 6 covers whole-number exponents (6.EE.A.1) and grade 8 covers the exponent rules with integer exponents (8.EE.A.1). In England, square and cube numbers come in Year 5, and wider work with powers in Key Stage 3.

Is there a free powers and exponents worksheet with answers?

Yes. Use the Print a worksheet button on this page. Each worksheet is freshly made, and the answer key can show answers only or answers with worked steps.

Grades 5 to 8 (Years 5 to 9 in England)

Exponents: powers, squares, cubes and the exponent rules

Here is what an exponent means, how to evaluate powers, squares and cubes, and why the exponent rules work, with examples you can check by writing the multiplication out. Try the free practice below, then print a fresh powers and exponents worksheet with an answer key.

The method, step by step

Read the power: in 2⁵, the base 2 is used as a factor 5 times.
To multiply powers with the same base, keep the base and add the exponents.
To divide powers with the same base, keep the base and subtract the exponents.
To raise a power to a power, keep the base and multiply the exponents.
Remember that any non-zero number to the power 0 is 1.
If you are ever unsure, write the multiplication out and count the factors.

CCSS 6.EE.A.1CCSS 8.EE.A.1England: KS3 NumberEngland: GCSE Foundation NumberEngland: Y6 Number – multiplication and division

Worked example

Simplify 3⁴ × 3² and find its value.

Both powers have the same base, 3, so add the exponents: 4 + 2 = 6.
3⁴ × 3² = 3⁶.
Check by writing it out: (3 × 3 × 3 × 3) × (3 × 3) is six 3s multiplied together.
Work up the powers of 3: 3² = 9, 3³ = 27, 3⁴ = 81, 3⁵ = 243, 3⁶ = 729.
Another check: 3⁴ × 3² = 81 × 9 = 729.
3⁴ × 3² = 3⁶ = 729.

Practice now

Question 1 of 100 correct

Write 4⁴ as a power of 2. What is the exponent?

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What an exponent means

An exponent is a short way to write repeated multiplication. In 5³, the 5 is the base and the small raised 3 is the exponent, also called the power or index. It means 5 × 5 × 5 = 125. You read it as 5 cubed, or 5 to the power 3.

The exponent counts the factors; it is not a number to multiply by. 2⁴ is 2 × 2 × 2 × 2 = 16, not 2 × 4 = 8. That one idea is behind every exponent rule on this page.

Powers of 10 show why exponents are useful. 10² = 100, 10³ = 1,000 and 10⁶ = 1,000,000: the exponent tells you how many zeros follow the 1.

Squares and cubes

A number to the power 2 is called squared, because n² is the area of a square with sides n long. A 7 cm by 7 cm square covers 7² = 49 cm². A number to the power 3 is called cubed, because n³ is the volume of a cube: a cube with edges 4 cm long has a volume of 4³ = 64 cm³.

Square numbers worth knowing: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.
Cube numbers worth knowing: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1,000.
Powers come before other operations: 6² + 2³ = 36 + 8 = 44, and 2 × 5² = 2 × 25 = 50.
Working back from a square gives a square root: a square patio with an area of 81 m² has sides of 9 m, because 9² = 81.

The exponent rules, and why they work

Every exponent rule comes from counting factors, and each one only works when the bases are the same.

Multiplying: add the exponents. 2³ × 2⁴ is (2 × 2 × 2) × (2 × 2 × 2 × 2), which is seven 2s, so 2³ × 2⁴ = 2⁷ = 128.
Dividing: subtract the exponents. In 5⁶ ÷ 5⁴, four 5s cancel from the top and bottom, leaving 5² = 25.
Power of a power: multiply the exponents. (4²)³ is 4² × 4² × 4², which is 4⁶, or 4,096.
Zero exponent: 7³ ÷ 7³ = 1, and the dividing rule gives 7⁰, so 7⁰ = 1.
Negative exponent: 2³ ÷ 2⁵ = 2⁻², which means 1/2² = 1/4.

The rules don't work across different bases, or across addition. 2³ × 3² cannot be written as one power; work it out as 8 × 9 = 72. And 2³ + 2⁴ is 8 + 16 = 24, not 2⁷.

Exponents with letters

The same rules work with variables. x⁵ × x³ = x⁸, y⁹ ÷ y⁴ = y⁵ and (a²)⁴ = a⁸. Numbers in front are dealt with separately: 3x² × 4x⁵ = 12x⁷, because 3 × 4 = 12 and 2 + 5 = 7. When a bracket is raised to a power, everything inside it is: (2x³)² = 4x⁶.

How it is taught in school

In the US, Common Core first uses whole-number exponents for powers of 10 in grade 5 (5.NBT.A.2). Grade 6 students write and evaluate expressions with whole-number exponents (6.EE.A.1), and grade 8 students apply the properties of integer exponents, which include zero and negative exponents (8.EE.A.1).

In England, Year 5 pupils learn square and cube numbers and the ² and ³ notation. Key Stage 3 extends this to integer powers and their roots. In UK classrooms exponents are usually called powers or indices, so the rules on this page are often called the laws of indices.

How parents can help

Learn the square numbers up to 12² together. They come up again in area, square roots and the Pythagorean theorem.
When a rule is forgotten, write the multiplication out. Counting factors rebuilds the rule in a few seconds.
Ask why 2⁵ is not 10. Explaining the difference between the base and the exponent fixes the most common mistake.
Spot powers of 10 in real life: a million is 10⁶ and a billion is 10⁹.
Keep sessions short. Print a fresh worksheet, let your child mark it with the answer key, and read the worked steps together for any they missed.

Common mistakes

Multiplying the base by the exponent. 3⁴ is 3 × 3 × 3 × 3 = 81, not 3 × 4 = 12.
Multiplying the bases when multiplying powers. 2³ × 2⁴ = 2⁷, not 4⁷: the base stays the same.
Adding exponents when the bases are different. 2³ × 5² is 8 × 25 = 200, and it cannot be written as a single power.
Adding instead of multiplying for a power of a power. (x²)³ = x⁶, not x⁵.
Thinking 5⁰ = 0. Any non-zero number to the power 0 is 1.
Forgetting the number in front when a bracket is raised to a power. (3x)² = 9x², not 3x².

Questions people ask

What are the exponent rules?: With the same base: multiply powers by adding the exponents, divide them by subtracting the exponents, and raise a power to a power by multiplying the exponents. Any non-zero number to the power 0 is 1, and a negative exponent means one over the positive power.
What is the difference between 3² and 3 × 2?: 3² means 3 × 3, which is 9. 3 × 2 is 6. The exponent tells you how many times the base is used as a factor.
Why is any number to the power 0 equal to 1?: Dividing a power by itself gives 1, and the dividing rule gives an exponent of 0. For example, 4³ ÷ 4³ = 64 ÷ 64 = 1, and the rule says 4³ ÷ 4³ = 4⁰, so 4⁰ = 1.
What are square and cube numbers?: A square number is a whole number multiplied by itself, such as 6² = 36. A cube number is a whole number used as a factor three times, such as 6³ = 216.
What does a negative exponent mean?: One over the positive power. 10⁻² = 1/10² = 1/100.
What grade learns exponents?: In the US, grade 6 covers whole-number exponents (6.EE.A.1) and grade 8 covers the exponent rules with integer exponents (8.EE.A.1). In England, square and cube numbers come in Year 5, and wider work with powers in Key Stage 3.
Is there a free powers and exponents worksheet with answers?: Yes. Use the Print a worksheet button on this page. Each worksheet is freshly made, and the answer key can show answers only or answers with worked steps.

Updated November 1, 2026.

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