What is the power of a power rule?

The power of a power rule says that when a power is raised to another power, you multiply the exponents: (aᵐ)ⁿ = a^(m × n). For example, (5²)³ means three copies of 5² multiplied together, 5² × 5² × 5². Adding 2 three times gives 6, and 2 × 3 is the quick way there, so the result is 5⁶.

How do you simplify exponents with the same base?

Check the bases first, because these laws only work when they match. Then read the sign between the powers: × means add the exponents, ÷ means subtract them, and an exponent outside a bracket means multiply. With different bases, such as 2³ × 3², no law combines them, so you work out each power instead: 8 × 9 = 72.

At what age do students learn the laws of exponents?

In many school systems, students meet these laws between the ages of about 11 and 14, once squares and cubes feel familiar. Both levels here are set for age 13. The year matters less than the habit behind it: a learner who can rebuild each law from repeated multiplication never has to memorise it.

Why does my child add the exponents in (2³)⁴?

The product rule usually comes first, so any two exponents start to look like a sum. Writing the bracket out fixes it: (2³)⁴ is 2³ × 2³ × 2³ × 2³, four groups of three 2s, which makes twelve 2s. So the answer is 2¹², not 2⁷. Real numbers confirm it: 8⁴ = 4096, and 2¹² = 4096 as well.

Can my students play this exponent laws game for free in a lesson?

They can: open the page in any browser on the class laptops or tablets, and both levels are free, with no logins or downloads. To start with some laws of exponents practice, write 3¹² in the middle of the board. Pupils then reach it with each law, such as 3⁵ × 3⁷, 3¹⁵ ÷ 3³ or (3⁴)³. On the schools plan, billed per student, a teacher dashboard shows who is getting quicker with exponents.

Math practice · reimagined

Exponent Rules Game

This exponent rules challenge never asks for a power’s value: the base stays between 2 and 9, and you type only the new exponent, from 1 to 36. Juggle those raised numbers faster than your rival, and your fighter throws the punches.

Ages 4–99+ · no download · plays in your browser · 23 languages

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What are the laws of exponents?

The laws of exponents are shortcuts for combining powers with the same base, the number that gets multiplied. Multiplying such powers adds the exponents, dividing subtracts them, and a power of a power multiplies them. Powers of 10 show it clearly: 10³ × 10² = 1000 × 100 = 100000, a 1 with five zeros, so it equals 10⁵.

Every level of Exponent laws

Example: what power of 8 is 8⁷ ÷ 8⁴?

Both powers have the base 8, so the answer is a power of 8 as well, and only its exponent is missing.
Write it as a fraction: seven 8s multiplied on top, four below. Each 8 below cancels one on top, so three 8s are left.
That is the subtraction 7 − 4 = 3, so 8⁷ ÷ 8⁴ = 8³.
Type 3, the exponent, on the keypad, not 512, which is the value of 8³.

Learn it step by step

When two powers share the same base, you don't have to work them out the long way — there's a shortcut that works on the exponents alone.

Multiplying ADDS the exponents: 2³ × 2⁵ means (2×2×2) × (2×2×2×2×2) — eight 2s in a row — so it's 2⁸, and 3 + 5 = 8. Dividing does the opposite and SUBTRACTS: 2⁷ ÷ 2² = 2⁵, because 7 − 2 = 5. The base never changes; only the exponent does.

Add (for ×) or subtract (for ÷) the exponents, and type the new one.

Tips and common mistakes

Each law drops one step down the ladder of operations: multiplying powers means adding exponents, dividing means subtracting, and a power of a power means multiplying.
A common mistake is multiplying the bases as well: 3² × 3⁴ is 3⁶, not 9⁶. Check with values: 9 × 81 = 729, and 3⁶ is 729 too.
Card game for home: draw two exponents, say 4 and 6, then race to give the new exponent for ×, ÷ and power of a power: 10, 2, 24.

How to play

Each round shows powers of one base, such as 6⁹ × 6² = 6^? or (7²)⁵ = 7^?, and a raised question mark marks the missing exponent. Type only that exponent on the keypad, here 11 or 10, never the huge value. Get it right first and Tomato attacks, with a blow that grows stronger the sooner you answer.

— Quick answers —

Exponent laws: the short FAQ

What is the power of a power rule?: The power of a power rule says that when a power is raised to another power, you multiply the exponents: (a^m)ⁿ = a^{(m × n)}. For example, (5²)³ means three copies of 5² multiplied together, 5² × 5² × 5². Adding 2 three times gives 6, and 2 × 3 is the quick way there, so the result is 5⁶.
How do you simplify exponents with the same base?: Check the bases first, because these laws only work when they match. Then read the sign between the powers: × means add the exponents, ÷ means subtract them, and an exponent outside a bracket means multiply. With different bases, such as 2³ × 3², no law combines them, so you work out each power instead: 8 × 9 = 72.
At what age do students learn the laws of exponents?: In many school systems, students meet these laws between the ages of about 11 and 14, once squares and cubes feel familiar. Both levels here are set for age 13. The year matters less than the habit behind it: a learner who can rebuild each law from repeated multiplication never has to memorise it.
Why does my child add the exponents in (2³)⁴?: The product rule usually comes first, so any two exponents start to look like a sum. Writing the bracket out fixes it: (2³)⁴ is 2³ × 2³ × 2³ × 2³, four groups of three 2s, which makes twelve 2s. So the answer is 2¹², not 2⁷. Real numbers confirm it: 8⁴ = 4096, and 2¹² = 4096 as well.
Can my students play this exponent laws game for free in a lesson?: They can: open the page in any browser on the class laptops or tablets, and both levels are free, with no logins or downloads. To start with some laws of exponents practice, write 3¹² in the middle of the board. Pupils then reach it with each law, such as 3⁵ × 3⁷, 3¹⁵ ÷ 3³ or (3⁴)³. On the schools plan, billed per student, a teacher dashboard shows who is getting quicker with exponents.

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