The laws of exponents are short rules for multiplying, dividing and raising powers to further powers.
2³ × 2² means multiplying by 2 three times and then twice more, which is 2 multiplied five times, 2⁵; so to multiply powers with the same base, you add the exponents.
In the same way, dividing subtracts the exponents, and raising a power to a power multiplies them.
On this page you change the base and exponent to see how fast powers grow, and count blocks to see why exponents add.
Multiplying the same number again and again
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Bar chart of how powers grow with the base and exponent
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👀 Powers
①2³ = 2 × 2 × 2 = 8: 2 multiplied three times, where 2 is the base and 3 the exponent
②Each time the exponent goes up by 1, the value is multiplied by the base again. That is why the bars in the picture climb more and more steeply
③The rules for multiplying the same number repeatedly are the laws of exponents
Law 1. Multiplying powers with the same base
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Counting a-blocks to see aᵐ × aⁿ = aᵐ⁺ⁿ
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Law 1. Multiplication
am × an = am+n
To multiply powers with the same base, add the exponents. m factors of a followed by n more make m + n factors
Laws 2 and 3. Division, and powers of powers
Law 2. Division
am ÷ an = am-n (m > n, a ≠ 0)
With the same base, subtract the exponents. If m = n the result is 1; if m < n it is 1 divided by a to the power (n − m)
Law 3. A power of a power
(am)n = amn
Raising a power to a power multiplies the exponents
💡 Why multiply the exponents?
①(a²)³ = a² × a² × a²: three groups of two a's, so a is multiplied 2 × 3 = 6 times
②In general, (aᵐ)ⁿ repeats "multiply by a, m times" n times, giving aᵐⁿ
③It is easy to mix this up with Law 1: multiplying powers adds exponents, but a power of a power multiplies them
Laws 4 and 5. Powers of products and quotients
Power of a product
(ab)n = an × bn
Raise each factor of the product to the power, including the number in front
Power of a quotient
(ab)n = anbn
Raise the numerator and the denominator to the power separately (b ≠ 0)
Check with examples
Example 1
Simplify a4 × a3.
1
To multiply powers with the same base, add the exponents.
a4 × a3 = a4+3
2
Work out the exponent.
= a7
▸ a7
Set m = 4 and n = 3 in the block picture: 4 a-blocks and 3 a-blocks make 7 in all.
Example 2
Simplify (2a2 b)3.
1
In a power of a product, raise each factor to the power, including the 2.
= 23 × (a2)3 × b3
2
A power of a power multiplies the exponents.
= 8a6 b3
▸ 8a6 b3
The most common slip is to forget to cube the 2 and write 2a6 b3.
Exam checklist
The laws of exponents
am·an=am+n, am÷an=am-n, (am)n=amn
am × an = am+n, am ÷ an = am−n, (am)n = amn, (ab)n = an bn
Grade 8 school test style
If an ÷ a2 = a5, what is the natural number n? (a ≠ 0)
①5
②6
③7
④8
⑤10
▸ ③ 7
1
To divide powers with the same base, subtract the exponents.
an ÷ a2 = an−2 = a5
2
Compare the exponents.
n − 2 = 5 ⇒ n = 7
🎯 Exam checklist
①First check the bases are the same; only then can you work with the exponents
②Multiplication adds exponents, division subtracts them, and a power of a power multiplies them
③In division, equal exponents give 1, and a larger exponent underneath leaves a fraction
④(−2)² = 4 but −2² = −4: brackets change the result completely
⑤In a power of a product, the number in front is raised to the power too: (2a)³ = 8a³