Everything covered in this lesson, in one place - useful for revision or printing.
\(a^n = \underbrace{a \times a \times \cdots \times a}_{n\ \text{times}}\)
An exponent is just a shortcut for repeated multiplication. The base \(a\) is what we multiply, and the power \(n\) tells us how many times.
\(x^4 \times x^5\)
Two powers being multiplied, with the same base \(x\). What happens to the exponents?
\((x\,x\,x\,x)(x\,x\,x\,x\,x)\)
Write each one out: that is four \(x\)’s times five \(x\)’s — nine \(x\)’s in total.
\(x^{4+5}\)
So we just add the exponents: \(4+5\).
\(\begin{gathered} x^4 \times x^5 = x^{9} \\[4pt] a^m \times a^n = a^{m+n}\end{gathered}\)
Product Law: same base → keep the base and add the powers.
\(\dfrac{x^6}{x^2}\)
Now we are dividing powers of the same base.
\(\dfrac{x\,x\,x\,x\,x\,x}{x\,x}\)
Two \(x\)’s on the bottom cancel two on the top, leaving four behind.
\(x^{6-2}\)
So we subtract the exponents: \(6-2\).
\(\begin{gathered} \dfrac{x^6}{x^2} = x^{4} \\[4pt] \dfrac{a^m}{a^n}=a^{m-n}\end{gathered}\)
Quotient Law: same base → subtract the bottom power from the top.
\((x^3)^2\)
A power raised to another power.
\(x^3 \times x^3\)
\((x^3)^2\) means \(x^3\) written twice and multiplied.
\(x^{3\times 2}\)
Using the Product Law that is \(x^{3+3}\) — or simply multiply the powers: \(3\times 2\).
\(\begin{gathered} (x^3)^2 = x^{6} \\[4pt] (a^m)^n = a^{mn}\end{gathered}\)
Power of a Power: multiply the exponents together.
\(\dfrac{x^3}{x^3}\)
Anything divided by itself equals \(1\). But let’s use the Quotient Law too…
\(x^{3-3} = x^{0}\)
The Quotient Law gives \(x^{3-3}=x^0\). Both answers must be equal!
\(x^{0}=1\qquad a^{0}=1\)
Zero Exponent: any non-zero base to the power \(0\) is \(1\) (\(a\neq 0\)).
\(\dfrac{x^2}{x^5}\)
Here the bigger power is on the bottom. Watch what the laws give us.
\(x^{2-5}=x^{-3}\quad\text{and}\quad \dfrac{1}{x^{3}}\)
The Quotient Law gives \(x^{-3}\); cancelling by hand gives \(\dfrac{1}{x^3}\). They must match.
\(x^{-3}=\dfrac{1}{x^{3}}\qquad a^{-n}=\dfrac{1}{a^{n}}\)
Negative Exponent: a negative power means take the reciprocal — it is not a negative number.
\((xy)^3\)
A whole product inside the bracket is raised to a power.
\((xy)(xy)(xy)\)
Three brackets multiplied. Regroup the \(x\)’s and \(y\)’s together.
\(x^{3}y^{3}\qquad (ab)^n = a^{n}b^{n}\)
Power of a Product: give the power to each factor.
\(\left(\dfrac{x}{y}\right)^3\)
Same idea, but now a fraction is raised to a power.
\(\dfrac{x}{y}\cdot\dfrac{x}{y}\cdot\dfrac{x}{y}\)
Three copies multiplied — the tops multiply together and the bottoms multiply together.
\(\dfrac{x^{3}}{y^{3}}\qquad \left(\dfrac{a}{b}\right)^n=\dfrac{a^{n}}{b^{n}}\)
Power of a Quotient: give the power to the top and the bottom.
\(\begin{gathered}\begin{aligned} a^m a^n &= a^{m+n} & (a^m)^n &= a^{mn} \\ \tfrac{a^m}{a^n} &= a^{m-n} & a^0 &= 1 \\ a^{-n} &= \tfrac{1}{a^n} & (ab)^n &= a^n b^n \end{aligned}\\[6pt] \left(\tfrac{a}{b}\right)^n=\tfrac{a^n}{b^n} \end{gathered}\)
You now know all seven laws. Same base for \(\times\) and \(\div\); multiply for powers of powers; spread the power across products and quotients. Ready for the quiz?