Laws of Exponents
Derive and apply the product rule, quotient rule, and power rule for exponents. Simplify expressions with integer exponents systematically, showing all steps and justifying each using the definition of exponentiation.
A lesson of Algebra 1, in Exponents & Exponential Functions, about 30 minutes.
By the end of this lesson, you will be able to multiply, divide and raise powers using the three laws, derive each of them from the definition when you forget, and avoid the two errors that account for nearly all lost marks here.
Everything Comes From Counting
$$x^5 = x \cdot x \cdot x \cdot x \cdot x$$
An exponent is a count of how many factors. Every rule in this lesson is that count being added, subtracted or multiplied, which means you never have to remember any of them: write the factors out and read off what happened.
$$x^2 \cdot x^3 = (x \cdot x)(x \cdot x \cdot x) = x^5$$
Two factors then three factors is five factors, so the exponents ADDED. That is the product rule, derived rather than memorized, and the same expansion settles every rule below.
Two words to keep straight. In $3x^4$, the base is $x$ and the exponent is $4$. The $3$ is a coefficient and is not being raised to anything, which matters immediately.
$$3x^4 \neq (3x)^4$$
The first is $3 \cdot x \cdot x \cdot x \cdot x$. The second is $3x \cdot 3x \cdot 3x \cdot 3x = 81x^4$.
The Three Laws
$$x^m \cdot x^n = x^{m+n} \qquad \textbf{product: add}$$
$$\frac{x^m}{x^n} = x^{m-n} \qquad \textbf{quotient: subtract}$$
$$(x^m)^n = x^{mn} \qquad \textbf{power of a power: multiply}$$
Each one from the counting:
Product. $m$ factors alongside $n$ factors is $m + n$ factors.
Quotient. Dividing cancels pairs, so $\frac{x^5}{x^2}$ loses two from the top and leaves $x^3$.
Power of a power. $(x^2)^3$ is $x^2 \cdot x^2 \cdot x^2$, which is three groups of two factors, so $6$ altogether. Groups of a size is multiplication.
The rules need the SAME BASE. $x^2 \cdot y^3$ does not combine at all, any more than $2x + 3y$ does, because there is nothing shared to count.
Sort It: Add, Subtract or Multiply
The fourth is worth pausing on. $x^2 \cdot x^2$ is a PRODUCT, so the exponents add to give $x^4$. Multiplying them would give $x^4$ as well, purely because $2+2$ and $2 \times 2$ both make $4$. That coincidence is exactly why the rule has to come from the operation rather than from the numbers in front of you.
Example 1: One of Each
Here is every step, worked all the way through. $$\frac{(x^3)^2 \cdot x^4}{x^5}$$
- Deal with the bracket first. What is (x^3)^2? x^6
- Now the top is x^6 times x^4. What is that? x^{10}
- Divide by x^5. What is the final answer? x^5
- How many factors of x is that, written out? 5