Powers
Definition
If $a$ is a real number, called the base, and $n$ is an integer, called the exponent, we define
with $a \in \mathbb{R}$ and $n \in \mathbb{Z}$.
| Property | Expression | Example |
|---|---|---|
| Product of powers with the same base | $a^{n} \cdot a^{m} = a^{n+m}$ | $(-2)^{2} \cdot (-2)^{3} = (-2)^{5} = -32$ |
| Quotient of powers with the same base | $\dfrac{a^{n}}{a^{m}} = a^{n-m}$ | $\dfrac{2^{8}}{2^{3}} = 2^{5} = 32$ |
| Power of a power | $(a^{n})^{m} = a^{n \cdot m}$ | $\left[\left(\tfrac{2}{3}\right)^{5}\right]^{2} = \left(\tfrac{2}{3}\right)^{10}$ |
| Power of a product | $(a \cdot b)^{n} = a^{n} \cdot b^{n}$ | $(2x)^{3} = 2^{3} x^{3} = 8x^{3}$ |
| Power of a quotient | $\left(\dfrac{a}{b}\right)^{n} = \dfrac{a^{n}}{b^{n}}$ | $\left(\dfrac{x}{2y}\right)^{5} = \dfrac{x^{5}}{2^{5} y^{5}}$ |
| Negative exponent | $a^{-n} = \dfrac{1}{a^{n}}$ | $(-2)^{-3} = \dfrac{1}{(-2)^{3}} = -\dfrac{1}{8}$ |
| Fractional exponent | $a^{p/q} = \sqrt[q]{a^{p}}$ | $3^{5/2} = \sqrt{3^{5}}$ |
| Zero exponent | $a^{0} = 1$ | $(-2xc)^{0} = 1$ |
| $(-a)^{n} \neq -a^{n}$ | $(-a)^{n} = \begin{cases} a^{n} & \text{if } n \text{ is even} \\ -a^{n} & \text{if } n \text{ is odd}\end{cases}$ | $(-2)^{2} = 4$ $(-2)^{3} = -8$ |
The one everybody trips on
The minus sign only goes into the power when it sits inside the brackets:
Negative exponents and fractions
A negative exponent flips the fraction:
A whole number works the same way, because $5 = \dfrac{5}{1}$:
Radicals
Definition
The $n$-th root of $A$ is the number $B$ such that $B^{n} = A$:
$A$ is the radicand, $n$ is the index of the root, and the whole expression $\sqrt[n]{A}$ is called a radical.
In the table below, $a$ and $b$ are positive. Without that condition the product and quotient rules stop holding.
| Property | Expression | Example |
|---|---|---|
| Product of roots with the same index | $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}$ | $\sqrt[3]{3} \cdot \sqrt[3]{9} = \sqrt[3]{27} = 3$ |
| Quotient of roots with the same index | $\dfrac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\dfrac{a}{b}}$ | $\dfrac{\sqrt[3]{16}}{\sqrt[3]{2}} = \sqrt[3]{8} = 2$ |
| Power of a root | $\left(\sqrt[n]{a}\right)^{p} = \sqrt[n]{a^{p}}$ | $\left(\sqrt[3]{x}\right)^{2} = \sqrt[3]{x^{2}}$ |
| Root of a root | $\sqrt[n]{\sqrt[m]{a}} = \sqrt[n \cdot m]{a}$ | $\sqrt[3]{\sqrt{x}} = \sqrt[6]{x}$ |
| Root of a product | $\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}$ | $\sqrt[3]{x^{2} y} = \sqrt[3]{x^{2}} \cdot \sqrt[3]{y}$ |
| Root of a quotient | $\sqrt[n]{\dfrac{a}{b}} = \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}$ | $\sqrt[3]{\dfrac{xb}{z}} = \dfrac{\sqrt[3]{x} \cdot \sqrt[3]{b}}{\sqrt[3]{z}}$ |
| Simplifying the radicand with $p = n \cdot c + r$ | $\sqrt[n]{a^{p}} = a^{c} \cdot \sqrt[n]{a^{r}}$ | $\sqrt[3]{x^{14}} = \sqrt[3]{x^{12+2}} = x^{4} \cdot \sqrt[3]{x^{2}}$ |
| Simplifying the index | $\sqrt[k \cdot n]{a^{k \cdot p}} = \sqrt[n]{a^{p}}$ | $\sqrt[12]{b^{8}} = \sqrt[3]{b^{2}}$ |
| Taking factors into the radicand | $a^{p} \cdot \sqrt[n]{b} = \sqrt[n]{b \cdot a^{n \cdot p}}$ | $m^{7} \cdot \sqrt[4]{m^{3}} = \sqrt[4]{(m^{7})^{4} \cdot m^{3}} = \sqrt[4]{m^{31}}$ |
| Changing the index | $\sqrt[n]{a^{p}} = \sqrt[n \cdot k]{a^{p \cdot k}}$ | $\sqrt[3]{x} = \sqrt[30]{x^{10}}$ |
| Sum of roots with the same index | $\sqrt[n]{a} + \sqrt[n]{b} \neq \sqrt[n]{a + b}$ | $\sqrt[3]{1} + \sqrt[3]{1} \neq \sqrt[3]{2}$ |
Why they have to be positive
The product rule comes from the fractional exponent:
Applied carelessly to negatives with an even index it produces nonsense: $\sqrt{6}$ is not $\sqrt{-2} \cdot \sqrt{-3}$, because $\sqrt{-2}$ and $\sqrt{-3}$ do not exist in $\mathbb{R}$.
Examples
With an odd index the root of a negative number does exist, and the minus sign comes out in front.
Left open in class, now solved
The first one calls for changing the index: the indices are different, $3$ and $4$, and both are turned into $12$, the least common multiple.
The second one is the power of a product read backwards: the exponents match, so the bases multiply.