Laws of Exponents

Preliminary Definitions

Natural Exponent

$$\boxed{a^n = \begin{cases} a & \text{if } n = 1 \\ \underbrace{a \cdot a \cdot \ldots \cdot a}_{n \text{ times}} & \text{if } n \in \mathbb{N}, n \geq 2 \end{cases}}$$

Sign rule for powers with a negative base:

$$(-b)^{2n} = +b^{2n}; \, \forall n \in \mathbb{Z}^+$$

$$(-b)^{2n+1} = -b^{2n+1}; \, \forall n \in \mathbb{Z}^+$$

Examples

• $3^4 = \underbrace{3 \cdot 3 \cdot 3 \cdot 3}_{4 \text{ times}} = 81$
• $\left(\dfrac{1}{2}\right)^3 = \underbrace{\dfrac{1}{2} \cdot \dfrac{1}{2} \cdot \dfrac{1}{2}}_{3 \text{ times}} = \dfrac{1}{8}$
• $(-2)^5 = \underbrace{(-2) \cdot (-2) \cdot (-2) \cdot (-2) \cdot (-2)}_{5 \text{ times}} = -32$
• $x^3 = \underbrace{x \cdot x \cdot x}_{3 \text{ times}}$
• $7^1 = 7 \quad \text{(by definition, if } n = 1\text{)}$
• $(-1)^6 = \underbrace{(-1) \cdot (-1) \cdot (-1) \cdot (-1) \cdot (-1) \cdot (-1)}_{6 \text{ times}} = 1$

Zero Exponent

$$\boxed{a^0 = 1}$$

$$\forall a \in \mathbb{R} \setminus \{0\}$$

Examples

• $5^0 = 1$
• $(-7)^0 = 1$
• $(3.14)^0 = 1$
• $\left(\frac{2}{3}\right)^0 = 1$
• $(x^2 + 1)^0 = 1 \quad \text{for any } x \in \mathbb{R}$
• $(a \cdot b \cdot c)^0 = 1 \quad \text{if } a, b, c \neq 0$

Negative Exponent

$$\boxed{a^{-n} = \frac{1}{a^n}}$$

$$\forall a \in \mathbb{R} \setminus \{0\}, n \in \mathbb{N}$$

Examples

• $2^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8}$
• $5^{-2} = \dfrac{1}{5^2} = \dfrac{1}{25}$
• $(-4)^{-3} = \dfrac{1}{(-4)^3} = \dfrac{1}{-64} = -\dfrac{1}{64}$
• $\left(\dfrac{1}{3}\right)^{-2} = \dfrac{1}{\left(\dfrac{1}{3}\right)^2} = \dfrac{1}{\dfrac{1}{9}} = 9$
• $x^{-4} = \dfrac{1}{x^4} \quad \text{if } x \neq 0$
• $(2a)^{-1} = \dfrac{1}{2a} \quad \text{if } a \neq 0$

Fractional Exponent

$$\boxed{a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m}$$

$$\forall n \in \mathbb{N}, n \geq 2$$

Examples

• $8^{2/3} = \sqrt[3]{8^2} = \sqrt[3]{64} = 4$
• $16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8$
• $27^{1/3} = \sqrt[3]{27} = 3$
• $4^{3/2} = (\sqrt{4})^3 = 2^3 = 8 \quad \text{(here } \sqrt[2]{\ } = \sqrt{\ }\text{)}$
• $x^{5/2} = \sqrt{x^5} = (\sqrt{x})^5 \quad \text{if } x \geq 0$
• $\left(\dfrac{1}{9}\right)^{1/2} = \sqrt{\dfrac{1}{9}} = \dfrac{1}{3}$

Exponentiation

Properties

Product of equal bases

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

$$a \in \mathbb{R}, m, n \in \mathbb{N}$$

Examples

• $2^3 \cdot 2^4 = 2^{3+4} = 2^7 = 128$
• $x^5 \cdot x^2 = x^{5+2} = x^7$
• $(-3)^2 \cdot (-3)^6 = (-3)^{2+6} = (-3)^8 = 6561$
• $a^4 \cdot a^1 = a^{4+1} = a^5$
• $\left(\dfrac{1}{2}\right)^3 \cdot \left(\dfrac{1}{2}\right)^5 = \left(\dfrac{1}{2}\right)^{3+5} = \left(\dfrac{1}{2}\right)^8 = \dfrac{1}{256}$
• $y^{n} \cdot y^{7} = y^{n+7}$

Power of a power

$$\boxed{(a^m)^n = a^{m \cdot n} \quad}$$

$$a \in \mathbb{R}, m, n \in \mathbb{N}$$

$$\boxed{\left(((a^m)^n)^r\right)^s = a^{m \cdot n \cdot r \cdot s}}$$

Examples

• $(2^3)^4 = 2^{3 \cdot 4} = 2^{12} = 4096$
• $(x^2)^5 = x^{2 \cdot 5} = x^{10}$
• $((-3)^4)^2 = (-3)^{4 \cdot 2} = (-3)^8 = 6561$
• $(a^3)^1 = a^{3 \cdot 1} = a^3$
• $\left(\left(\dfrac{1}{2}\right)^2\right)^3= \left(\dfrac{1}{2}\right)^{2 \cdot 3} = \left(\dfrac{1}{2}\right)^6 = \dfrac{1}{64}$
• $(y^m)^7 = y^{m \cdot 7} = y^{7m}$

Power of a product

$$\boxed{(a \cdot b)^n = a^n \cdot b^n}$$

$$a, b \in \mathbb{R}, n \in \mathbb{N}$$

$$\boxed{(x^a \cdot y^b)^n = x^{a \cdot n} \cdot y^{b \cdot n}}$$

Examples

• $(3 \cdot 4)^2 = 3^2 \cdot 4^2 = 9 \cdot 16 = 144$
• $(2x)^3 = 2^3 \cdot x^3 = 8x^3$
• $(-5 \cdot 2)^4 = (-5)^4 \cdot 2^4 = 625 \cdot 16 = 10000$
• $(ab)^5 = a^5 \cdot b^5$
• $\left(\dfrac{1}{2} \cdot y\right)^3 = \left(\dfrac{1}{2}\right)^3 \cdot y^3 = \dfrac{1}{8}y^3$
• $(xy^2)^4 = x^4 \cdot (y^2)^4 = x^4 \cdot y^8$

Division of equal bases

$$\boxed{\frac{a^m}{a^n} = a^{m-n}}$$

$$m, n \in \mathbb{N}, m \geq n, a \in \mathbb{R} \setminus \{0\}$$

Examples

• $\dfrac{5^7}{5^3} = 5^{7-3} = 5^4 = 625$
• $\dfrac{x^8}{x^5} = x^{8-5} = x^3$
• $\dfrac{(-2)^6}{(-2)^2} = (-2)^{6-2} = (-2)^4 = 16$
• $\dfrac{a^{10}}{a^{10}} = a^{10-10} = a^0 = 1 \quad \text{(if } a \neq 0\text{)}$
• $\dfrac{3^5}{3^2} = 3^{5-2} = 3^3 = 27$
• $\dfrac{y^{n+4}}{y^n} = y^{(n+4)-n} = y^4$

Power of a quotient

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

$$n \in \mathbb{N}, b \in \mathbb{R} \setminus \{0\}$$

Examples

• $\left(\dfrac{2}{3}\right)^4 = \dfrac{2^4}{3^4} = \dfrac{16}{81}$
• $\left(\dfrac{5}{2}\right)^3 = \dfrac{5^3}{2^3} = \dfrac{125}{8}$
• $\left(\dfrac{x}{4}\right)^2 = \dfrac{x^2}{4^2} = \dfrac{x^2}{16} \quad \text{if } x \in \mathbb{R}$
• $\left(\dfrac{-3}{5}\right)^3 = \dfrac{(-3)^3}{5^3} = \dfrac{-27}{125}$
• $\left(\dfrac{a}{b}\right)^5 = \dfrac{a^5}{b^5} \quad \text{if } b \neq 0$
• $\left(\dfrac{1}{x}\right)^n = \dfrac{1^n}{x^n} = \dfrac{1}{x^n} \quad \text{if } x \neq 0$

Negative exponent of a fraction

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

$$a, b \neq 0$$

Examples

• $\left(\dfrac{2}{3}\right)^{-4} = \left(\dfrac{3}{2}\right)^4 = \dfrac{3^4}{2^4} = \dfrac{81}{16}$
• $\left(\dfrac{5}{x}\right)^{-2} = \left(\dfrac{x}{5}\right)^2 = \dfrac{x^2}{25} \quad \text{if } x \neq 0$
• $\left(\dfrac{1}{4}\right)^{-3} = \left(\dfrac{4}{1}\right)^3 = 4^3 = 64$
• $\left(\dfrac{a}{b}\right)^{-1} = \dfrac{b}{a} \quad \text{if } a, b \neq 0$
• $\left(\dfrac{-2}{7}\right)^{-3} = \left(\dfrac{7}{-2}\right)^3 = \dfrac{7^3}{(-2)^3} = \dfrac{343}{-8} = -\dfrac{343}{8}$
• $\left(\dfrac{x}{y}\right)^{-5} = \dfrac{y^5}{x^5} \quad \text{if } x, y \neq 0$

Successive Exponents

$$\boxed{x^{a^{b^c}} = x^{a^m}= x^n= z}$$

Examples

• $2^{2^{3^1}} = 2^{2^3} = 2^8 = 256$
• $3^{2^{2^2}} = 3^{2^4} = 3^{16} = 43,\!046,\!721$
• $5^{1^{4^2}} = 5^{1^{16}} = 5^1 = 5$
• $10^{3^{1^5}} = 10^{3^1} = 10^3 = 1,\!000$
• $2^{3^{2^1}} = 2^{3^2} = 2^9 = 512$
• $4^{2^{3^0}} = 4^{2^1} = 4^2 = 16$

Absolute Value

$$\boxed{ \sqrt[2k]{x^{2k}} = |x| = \begin{cases} x & \text{if: } x > 0 \\ 0 & \text{if: } x = 0 \\ -x & \text{if: } x < 0 \end{cases}}$$

Radicals in $\mathbb{R}$

Sign rule:

$$\sqrt[2n+1]{(+)} = (+)$$

$$\sqrt[2n+1]{(-)} = (-)$$

$$\sqrt[2n]{(+)} = (+)$$

$$\sqrt[2n]{(-)} = \text{not defined in }\mathbb{R}$$

Properties

Root of a product

$$\boxed{\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}}$$

If $n$ is even, then $a \geq 0$ and $b \geq 0$.

Examples

• $\sqrt{4 \cdot 9} = \sqrt{4} \cdot \sqrt{9} = 2 \cdot 3 = 6$
• $\sqrt[3]{8 \cdot 27} = \sqrt[3]{8} \cdot \sqrt[3]{27} = 2 \cdot 3 = 6$
• $\sqrt[4]{16 \cdot 81} = \sqrt[4]{16} \cdot \sqrt[4]{81} = 2 \cdot 3 = 6$
• $\sqrt{x \cdot y} = \sqrt{x} \cdot \sqrt{y} \quad \text{if } x \geq 0,\ y \geq 0$
• $\sqrt[5]{a^5 \cdot b^{10}} = \sqrt[5]{a^5} \cdot \sqrt[5]{b^{10}} = a \cdot b^2$
• $\sqrt[3]{-8 \cdot 64} = \sqrt[3]{-8} \cdot \sqrt[3]{64} = (-2) \cdot 4 = -8$

Root of a quotient

$$\boxed{\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}}$$

$$b \neq 0$$

If $n$ is even, then $a \geq 0$ and $b > 0$.

Examples

• $\sqrt{\dfrac{9}{16}} = \dfrac{\sqrt{9}}{\sqrt{16}} = \dfrac{3}{4}$
• $\sqrt[3]{\dfrac{27}{125}} = \dfrac{\sqrt[3]{27}}{\sqrt[3]{125}} = \dfrac{3}{5}$
• $\sqrt[4]{\dfrac{1}{81}} = \dfrac{\sqrt[4]{1}}{\sqrt[4]{81}} = \dfrac{1}{3}$
• $\sqrt[5]{\dfrac{x^5}{32}} = \dfrac{\sqrt[5]{x^5}}{\sqrt[5]{32}} = \dfrac{x}{2} \quad \text{if } x \in \mathbb{R}$
• $\sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}} \quad \text{if } a \geq 0,\ b > 0$
• $\sqrt[3]{\dfrac{-8}{27}} = \dfrac{\sqrt[3]{-8}}{\sqrt[3]{27}} = \dfrac{-2}{3}$

Root of a root

$$\boxed{\sqrt[m]{\sqrt[n]{a}} = \sqrt[m \cdot n]{a}}$$

$$m, n \in \mathbb{N}$$

If $m \cdot n$ is even, then $a \geq 0$.

$$\boxed{\sqrt[m]{\sqrt[n]{\sqrt[s]{\sqrt[r]{a}}}} = \sqrt[m \cdot n \cdot s \cdot r]{a}}$$

Examples

• $\sqrt[2]{\sqrt[3]{8}} = \sqrt[6]{8} = \sqrt[6]{2^3} = 2^{3/6} = 2^{1/2} = \sqrt{2}$
• $\sqrt[3]{\sqrt[4]{x}} = \sqrt[12]{x} \quad \text{for } x \geq 0$
• $\sqrt{\sqrt{16}} = \sqrt[2]{\sqrt[2]{16}} = \sqrt[4]{16} = 2$
• $\sqrt[5]{\sqrt{a}} = \sqrt[10]{a} \quad \text{if } a \geq 0$
• $\sqrt[4]{\sqrt[3]{64}} = \sqrt[12]{64} = \sqrt[12]{2^6} = 2^{6/12} = 2^{1/2} = \sqrt{2}$
• $\sqrt{\sqrt{\sqrt{x}}} = \sqrt[2]{\sqrt[2]{\sqrt[2]{x}}} = \sqrt[8]{x} \quad \text{for } x \geq 0$

Power of a root

$$\boxed{\left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}}$$

$$\boxed{\sqrt[ab]{x^{ac}} = \sqrt[b]{x^c} \quad \text{If } ab \text{ is even, then } x \in \mathbb{R}_0^+}$$

Examples

• $\left(\sqrt[3]{2}\right)^6 = \sqrt[3]{2^6} = \sqrt[3]{64} = 4$
• $\left(\sqrt{5}\right)^4 = \sqrt{5^4} = \sqrt{625} = 25$
• $\left(\sqrt[4]{3}\right)^2 = \sqrt[4]{3^2} = \sqrt[4]{9}$
• $\left(\sqrt[n]{x}\right)^3 = \sqrt[n]{x^3} \quad \text{if } x \geq 0 \text{ when } n \text{ is even}$
• $\left(\sqrt[5]{-32}\right)^3 = \sqrt[5]{(-32)^3} = \sqrt[5]{-32768} = -8$
• $\left(\sqrt{a^2 + b^2}\right)^2 = \sqrt{(a^2 + b^2)^2} = |a^2 + b^2| = a^2 + b^2$

Successive Radicals

$$\sqrt[n]{x^{a}\cdot \sqrt[m]{x^{b}\cdot \sqrt[p]{x^c}}} = \sqrt[nmp]{x^{a \cdot m \cdot p + b \cdot p + c}}$$

$$\sqrt[n]{x^{a}\div \sqrt[m]{x^{b}\div \sqrt[p]{x^c}}} = \sqrt[nmp]{x^{a \cdot m \cdot p - b \cdot p + c}}$$

$$\sqrt[m]{x \cdot \sqrt[n]{y \cdot \sqrt[p]{z}}} = \sqrt[m]{x} \cdot \sqrt[m \cdot n]{y} \cdot \sqrt[m \cdot n \cdot p]{z}$$

$$\sqrt[m]{x^a \cdot \sqrt[n]{y^b \cdot \sqrt[p]{z^c}}} = \sqrt[m]{x^a} \cdot \sqrt[m \cdot n]{y^b} \cdot \sqrt[m \cdot n \cdot p]{z^c}$$

$$\sqrt[m]{x^{a}\div \sqrt[n]{y^{b}\div \sqrt[p]{z^{c}\div \sqrt[q]{w^{d}\div \sqrt[r]{v^{e}\div \sqrt[s]{u^f}}}}}} = \frac{x^{\frac{a}{m}} \cdot z^{\frac{c}{mnp}} \cdot v^{\frac{e}{mnpqr}} }{y^{\frac{b}{mn}} \cdot w^{\frac{d}{mnpq}} \cdot u^{\frac{f}{mnpqrs}}}$$

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Auxiliary Properties

$$x^m \cdot \sqrt[n]{y^p} = \sqrt[n]{x^{mn} \cdot y^p}$$

$$\sqrt[m]{x} \cdot \sqrt[n]{x} = \sqrt[mn]{x^{n+m}}$$

$$\frac{\sqrt[m]{x}}{\sqrt[n]{x}} = \sqrt[mn]{x^{n-m}}, \, x \neq 0$$

$$\sqrt[a]{\frac{\sqrt[b]{\cfrac{\sqrt[c]{\cfrac{1}{z}}}{y}}}{x}} = \sqrt[a]{\frac{1}{x}\cdot \sqrt[b]{\frac{1}{y}\cdot \sqrt[c]{\frac{1}{z}}}}$$

$$\frac{\sqrt[m]{a\cdot \sqrt[n]{b\cdot \sqrt[p]{c}} } }{\sqrt[m]{d\cdot \sqrt[n]{e\cdot \sqrt[p]{f}} } } = \sqrt[m]{\frac{a}{d}\cdot \sqrt[n]{\frac{b}{e}\cdot \sqrt[p]{\frac{c}{f}}}}$$

$$\sqrt[m]{x^{ \sqrt[n]{y^{ \sqrt[p]{z}}}}} = \sqrt[m]{x}^{\sqrt[n]{y}^{\sqrt[p]{z}}}$$

Special Cases

1. Expressions with a specific number of radicals

$$\underbrace{\sqrt[n]{ \sqrt[n]{ \sqrt[n]{ \dots \sqrt[n]{x}}}}}_{\text{m radicals}} = \sqrt[n^m]{x}$$

$$\underbrace{\sqrt[n]{x \cdot \sqrt[n]{x \cdot \sqrt[n]{x \cdot ...\cdot \sqrt[n]{x}}}}}_{\text{m radicals}} = \sqrt[n^m]{x^{\cfrac{n^m - 1}{n-1}}}$$

$$\underbrace{\sqrt[n]{x\div \sqrt[n]{x\div \sqrt[n]{x\div ... \cdot \div \sqrt[n]{x}}}}}_{''m'' \text{ radicals}} = \begin{cases} \sqrt[n^m]{x^{\cfrac{n^m - 1}{n + 1}}} & \text{If 'm' is even}, \\ \sqrt[n^m]{x^{\cfrac{n^m + 1}{n + 1}}} & \text{If 'm' is odd}. \end{cases}$$

2. Expressions with infinite radicals

$$\sqrt{x\cdot(x+1)+ \sqrt{x\cdot(x+1)+ \sqrt{x\cdot(x+1)+ \dots } }} = x+1$$

$$\sqrt{x\cdot(x+1)- \sqrt{x\cdot(x+1)- \sqrt{x\cdot(x+1)- \dots } }} = x$$

$$\sqrt[n]{x\cdot \sqrt[n]{x\cdot \sqrt[n]{x\cdot...} }} = \sqrt[n-1]{x}$$

$$\sqrt[n]{x\div \sqrt[n]{x\div \sqrt[n]{x\div...} }} = \sqrt[n+1]{x}$$

$$\quad \sqrt[n]{a \cdot \sqrt[m]{a \cdot \sqrt[n]{a \cdot \sqrt[m]{a\cdot \dots} }}} = \sqrt[m \cdot n - 1]{a^{m+1}}$$

$$\quad \sqrt[n]{a \cdot \sqrt[m]{b \cdot \sqrt[n]{a \cdot \sqrt[m]{b\cdot \dots} }}} = \sqrt[m \cdot n - 1]{a^m \cdot b}$$

$$\quad \sqrt{a + \sqrt{a + \sqrt{a + \dots}}} = \frac{1 + \sqrt{4a + 1}}{2}$$

$$\quad \sqrt{a - \sqrt{a - \sqrt{a - \dots}}} = \frac{-1 + \sqrt{4a + 1}}{2}$$

3. Continuous infinite exponential

$$x^{x^{x^{{.}^{{.}^{{.}^{x^n}}}}}} = n \quad \rightarrow \quad x = \sqrt[n]{n}; \, x \neq 0$$

$$\sqrt[a]{b}^{\sqrt[a]{b}^{\sqrt[a]{b}^{\sqrt[a]{b}^{.^{.^{.}}}}}} = c \quad \rightarrow \quad b = c^{\,a}$$