Laws of exponents

The definition and where the laws come from#

$a^n$ means that the factor $a$ is used $n$ times: $2^5 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32$. The number $a$ is the base, and $n$ is the exponent (or power).

Any law can be checked by writing the power out in full. For example, what is $2^3 \cdot 2^4$? Write it out: three twos, then four more twos, seven factors in all — so $2^7$. Hence the rule: the exponents add. What you need to remember is not the formula but how to get it: exponents get mixed up, but counting factors never lets you down.

The five laws#

When you multiply powers with the same base, add the exponents:

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

When you divide, subtract them:

$a^m \div a^n = a^{m-n}$

When you raise a power to a power, multiply them:

$(a^m)^n = a^{mn}$

A power of a product is the product of the powers:

$(ab)^n = a^n b^n$

A power of a fraction is a fraction of powers:

$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

The zero exponent stands apart: $a^0 = 1$ for any $a \ne 0$. This is not an arbitrary convention but a consequence of the division law: $a^3 \div a^3 = a^{3-3} = a^0$, and any number divided by itself equals one.

An important limit: the first three laws need the SAME base, and the last two the same exponent. For $2^3 \cdot 3^2$ there is no rule — you have to calculate: $8 \cdot 9 = 72$.

Worked examples#

Example 1. Work out $2^3 \cdot 2^4$.

Step 1. The bases are the same — both twos — so the first law applies.

Step 2. Add the exponents: $3 + 4 = 7$, which gives $2^7$.

Step 3. Find the value: $2^7 = 128$.

Multiplying the exponents would be wrong here: $2^{12}$ is 4096, which is 32 times the correct answer.

Example 2. Work out $3^7 \div 3^5$.

Step 1. Same bases, and the operation is division, so the exponents are subtracted.

Step 2. $7 - 5 = 2$, which gives $3^2$.

Step 3. $3^2 = 9$.

Example 3. Simplify $(2a^3)^2$.

Step 1. The bracket holds a product, so every factor is squared: $2^2 \cdot (a^3)^2$.

Step 2. Work out the number: $2^2 = 4$. People often leave it as 2 — that is the main mistake in this example.

Step 3. Power of a power: multiply the exponents, $(a^3)^2 = a^6$.

Answer: $4a^6$.

The minus sign and where it gets lost#

$(-2)^4$ and $-2^4$ mean different things. In the first, the whole $-2$ is raised to the fourth power: $(-2)^4 = 16$. In the second, 2 is raised to the power first and the minus stays outside: $-2^4 = -16$. The brackets change the answer, not just the look.

The sign rule for a negative base: an even exponent gives a plus, an odd one gives a minus. $(-2)^3 = -8$, $(-2)^4 = 16$, and $(-3)^5$ is negative. The reason is simple — minus signs cancel in pairs, and with an even exponent the pairs come out with nothing left over.

What you cannot do with powers#

Add powers using the multiplication law. There is no law for $a^3 + a^4$: it is a sum, not a product, and the most you can do is factor it — $a^3(1 + a)$.

Raise a sum to a power term by term. $(a+b)^2 = a^2 + b^2$ is false: there is a middle term, twice the product, between the squares — see special products in algebra.

Apply the base rules to different bases. By the power-of-a-product law, $2^3 \cdot 5^3 = (2 \cdot 5)^3 = 1000$: here the exponents match, not the bases, so the fourth law works, not the first.

Forget that anything to the power zero is one: $7^0 = 1$, not 0 and not 7.

Where you will use this straight away#

Squares and cubes appear in the formulas for area and volume: the side of a square squared gives its area — which is where the name of the power comes from; more in area and perimeter. Powers of ten turn long numbers into short ones: $10^3 = 1000$, $10^6$ is a million. And computer sizes are powers of two: $2^{10} = 1024$, which is why people say "a kilobyte is 1024 bytes". Strictly by the standard, 1024 bytes is a kibibyte and a kilobyte is 1000 bytes, but in everyday speech almost nobody says it that way.

You have the topic when, looking at an expression, you can immediately name the law out loud — "same base, multiplying, add the exponents" — and you do not slip on the sign with a negative base.

Step-by-step plan

1. Step 1 — a table of powersLearn the powers of two up to 2¹⁰, of three up to 3⁵ and the squares of numbers up to 20.
2. Step 2 — the first two lawsTen examples each of multiplying and dividing powers with the same base.
3. Step 3 — power of a power and of a productSimplify expressions like (3x²)³ without forgetting to raise the number as well.
4. Step 4 — signsTell (−a)ⁿ from −aⁿ in twenty examples with even and odd exponents.
5. Step 5 — mixed expressionsSimplify expressions where several laws apply in a row and check by substituting a = 2.

Check yourself

Sources

• Khan Academy — Pre-algebraExponent rules with practice and instant feedback
  free
• OpenStax Elementary Algebra 2eFree textbook; the polynomials chapter covers the properties of exponents
  free

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