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}}

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Note

The expression:

{\underbrace{(\sqrt{xy})\cdot(\sqrt{xy})\cdot(\sqrt{xy}) ... (\sqrt{xy})}_{(\sqrt{ 5 }+\sqrt{ 3 }) \text{ times}}\neq (\sqrt{xy})^{\sqrt{ 5 }+\sqrt{ 3 }}}

is not defined, because the exponent must be a natural number.

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}^+

In summary:

(-)^{\text{even}}=+

(-)^{\text{odd}}=-

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

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Do Not Forget

\underbrace{3 + 3 + 3 + \cdots + 3}_{10 \text{ times}} = 3 \cdot 10

\underbrace{5 + 5 + 5 + \cdots + 5}_{20 \text{ times}} = 5 \cdot 20

\underbrace{a + a + a + \cdots + a}_{15 \text{ times}} = 15 \cdot a

\underbrace{3 \cdot 3 \cdot 3 \cdot \cdots \cdot 3}_{10 \text{ times}} = 3^{10}

\underbrace{5 \cdot 5 \cdot 5 \cdot \cdots \cdot 5}_{20 \text{ times}} = 5^{20}

\underbrace{a \cdot a \cdot a \cdot \cdots \cdot a}_{15 \text{ times}} = a^{15}

Zero Exponent

\boxed{a^0 = 1}

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

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Note:

$0^0$ is an indeterminate form.

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

Negative Exponent

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

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

📋

Note:

$0^{-n}$ is not defined for $n \in \mathbb{N}$ .

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

Fractional Exponent

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

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

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

Exponentiation

Fundamental Identity

\boxed{P = a^n}

a \in \mathbb{R}, n \in \mathbb{N}, P \in \mathbb{R}

Where:

$a$ : base
$n$ : natural exponent
$P$ : power

Properties

Product of equal bases

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

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

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

Warning

a^5+a^3 \not= a^8

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}}

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Note:

a^{7a}=(a^a)^7

2^{2a}=(2^a)^2

(2^3)^2 \not =2^{3^2}

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

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}}

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

Warning

(a+b)^2 \not= a^2+b^2

5 \cdot 2^x \not= 10^x

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\}

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

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\}

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

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

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

Successive Exponents

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

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Note:

{(x^m)^n \neq x^{m^n}}

{(x+y)^n \neq x^n+y^n}

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

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}}

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Remember

Commonly used powers


$2^2 = 4$	$3^2 = 9$	$4^2 = 16$	$8^2 = 64$
$2^3 = 8$	$3^3 = 27$	$4^3 = 64$	$8^3 = 512$
$2^4 = 16$	$3^4 = 81$	$5^2 = 25$	$9^2 = 81$
$2^5 = 32$	$3^5 = 243$	$5^3 = 125$	$9^3 = 729$
$2^6 = 64$	$3^6 = 729$	$5^4 = 625$	$10^2 = 100$
$2^7 = 128$		$5^5 = 3125$	$10^3 = 1000$
$2^8 = 256$		$6^2 = 36$	$11^2 = 121$
$2^9 = 512$		$6^3 = 216$	$11^3 = 1331$
$2^{10} = 1024$		$6^4 = 1296$	$12^2 = 144$
		$7^2 = 49$	$12^3 = 1728$
		$7^3 = 343$
		$7^4 = 2401$

Radicals in $\mathbb{R}$

Fundamental Identity

\boxed{y = \sqrt[n]{x} \iff y^n = x }

Where:

$x$ : radicand ( $x \in \mathbb{R}, x \geq 0$ if $n$ is even)
$n$ : index ( $n \in \mathbb{N}, n \geq 2$ )
$y$ : root ( $y \geq 0$ if $n$ is even)

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Remember

Every root of zero is zero
(regardless of its index).

Examples:

$\sqrt{0} = 0$
$\sqrt[3]{0} = 0$
$\sqrt[5]{0} = 0$
$\sqrt[7]{0} = 0$

Every root of one is one
(regardless of its index).

Examples:

$\sqrt{1} = 1$
$\sqrt[3]{1} = 1$
$\sqrt[5]{1} = 1$
$\sqrt[7]{1} = 1$

Sign rule:

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

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

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

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

In summary:

\sqrt[\text{odd}]{+} = +

\sqrt[\text{odd}]{-} = -

\sqrt[\text{even} ]{+} = +

\sqrt[\text{even} ]{-} = \text{does not exist}

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 $b \geq 0$ .

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

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Remember

Square Roots (simplified forms)

$2\sqrt{2} =\sqrt{8}$
$3\sqrt{2} =\sqrt{18}$
$4\sqrt{2} =\sqrt{32}$
$5\sqrt{2} =\sqrt{50}$
$3\sqrt{3} =\sqrt{27}$
$4\sqrt{3} =\sqrt{48}$
$5\sqrt{3} =\sqrt{75}$
$3\sqrt{5} =\sqrt{45}$
$4\sqrt{5} =\sqrt{80}$
$2\sqrt{5} =\sqrt{20}$

Cube Roots (simplified forms)

$2\sqrt[3]{2} =\sqrt[3]{16}$
$3\sqrt[3]{2} =\sqrt[3]{54}$
$2\sqrt[3]{3} =\sqrt[3]{24}$
$3\sqrt[3]{3} =\sqrt[3]{81}$

Approximate decimal values (square roots)

$\sqrt{2} \approx 1.4142...$
$\sqrt{3} \approx 1.732...$
$\sqrt{5} \approx 2.236...$

Warning

\sqrt{a+b} \neq \sqrt{a}+\sqrt{b}

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 $b > 0$ .

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

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}}

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

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^+}

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

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}}}

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

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}

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}

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}

Laws of Exponents

Preliminary Definitions

Natural Exponent

Note

📋 Examples

Do Not Forget

Zero Exponent

Note:

📋 Examples

Negative Exponent

Note:

📋 Examples

Fractional Exponent

📋 Examples

Exponentiation

Fundamental Identity

Properties

Product of equal bases

📋 Examples

Warning

Power of a power

Note:

📋 Examples

Power of a product

📋 Examples

Warning

Division of equal bases

📋 Examples

Power of a quotient

📋 Examples

Negative exponent of a fraction

📋 Examples

Successive Exponents

Note:

📋 Examples

Absolute Value

Remember

Radicals in R\mathbb{R}R

Fundamental Identity

Remember

Properties

Root of a product

📋 Examples

Remember

Warning

Root of a quotient

📋 Examples

Root of a root

📋 Examples

Power of a root

📋 Examples

Successive Radicals

Auxiliary Properties

Special Cases

Radicals in $\mathbb{R}$