Laws of Exponents

Preliminary Definitions

Natural Exponent

an={aif n=1a⋅a⋅…⋅a⏟n timesif n∈N,n≥2\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=+b2n; ∀n∈Z+(-b)^{2n} = +b^{2n}; \, \forall n \in \mathbb{Z}^+

(−b)2n+1=−b2n+1; ∀n∈Z+(-b)^{2n+1} = -b^{2n+1}; \, \forall n \in \mathbb{Z}^+

In summary:

(−)even=+(-)^{\text{even}}=+

(−)odd=−(-)^{\text{odd}}=-

📝 Examples

  • 34=3⋅3⋅3⋅3⏟4 times=813^4 = \underbrace{3 \cdot 3 \cdot 3 \cdot 3}_{4 \text{ times}} = 81
  • (12)3=12⋅12⋅12⏟3 times=18\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=(−2)⋅(−2)⋅(−2)⋅(−2)⋅(−2)⏟5 times=−32(-2)^5 = \underbrace{(-2) \cdot (-2) \cdot (-2) \cdot (-2) \cdot (-2)}_{5 \text{ times}} = -32
  • x3=x⋅x⋅x⏟3 timesx^3 = \underbrace{x \cdot x \cdot x}_{3 \text{ times}}
  • 71=7(by definition, if n=1)7^1 = 7 \quad \text{(by definition, if } n = 1\text{)}
  • (−1)6=(−1)⋅(−1)⋅(−1)⋅(−1)⋅(−1)⋅(−1)⏟6 times=1(-1)^6 = \underbrace{(-1) \cdot (-1) \cdot (-1) \cdot (-1) \cdot (-1) \cdot (-1)}_{6 \text{ times}} = 1
🧠

Do Not Forget

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

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

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

3⋅3⋅3⋅⋯⋅3⏟10 times=310\underbrace{3 \cdot 3 \cdot 3 \cdot \cdots \cdot 3}_{10 \text{ times}} = 3^{10}

5⋅5⋅5⋅⋯⋅5⏟20 times=520\underbrace{5 \cdot 5 \cdot 5 \cdot \cdots \cdot 5}_{20 \text{ times}} = 5^{20}

a⋅a⋅a⋅⋯⋅a⏟15 times=a15\underbrace{a \cdot a \cdot a \cdot \cdots \cdot a}_{15 \text{ times}} = a^{15}

Zero Exponent

a0=1\boxed{a^0 = 1}

∀a∈R∖{0}\forall a \in \mathbb{R} \setminus \{0\}

📝 Examples

  • 50=15^0 = 1
  • (−7)0=1(-7)^0 = 1
  • (3.14)0=1(3.14)^0 = 1
  • (23)0=1\left(\frac{2}{3}\right)^0 = 1
  • (x2+1)0=1for any x∈R(x^2 + 1)^0 = 1 \quad \text{for any } x \in \mathbb{R}
  • (a⋅b⋅c)0=1if a,b,c≠0(a \cdot b \cdot c)^0 = 1 \quad \text{if } a, b, c \neq 0

Negative Exponent

a−n=1an\boxed{a^{-n} = \frac{1}{a^n}}

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

📝 Examples

  • 2−3=123=182^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8}
  • 5−2=152=1255^{-2} = \dfrac{1}{5^2} = \dfrac{1}{25}
  • (−4)−3=1(−4)3=1−64=−164(-4)^{-3} = \dfrac{1}{(-4)^3} = \dfrac{1}{-64} = -\dfrac{1}{64}
  • (13)−2=1(13)2=119=9\left(\dfrac{1}{3}\right)^{-2} = \dfrac{1}{\left(\dfrac{1}{3}\right)^2} = \dfrac{1}{\dfrac{1}{9}} = 9
  • x−4=1x4if x≠0x^{-4} = \dfrac{1}{x^4} \quad \text{if } x \neq 0
  • (2a)−1=12aif a≠0(2a)^{-1} = \dfrac{1}{2a} \quad \text{if } a \neq 0

Fractional Exponent

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

∀n∈N,n≥2\forall n \in \mathbb{N}, n \geq 2

📝 Examples

  • 82/3=823=643=48^{2/3} = \sqrt[3]{8^2} = \sqrt[3]{64} = 4
  • 163/4=(164)3=23=816^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8
  • 271/3=273=327^{1/3} = \sqrt[3]{27} = 3
  • 43/2=(4)3=23=8(here  2= )4^{3/2} = (\sqrt{4})^3 = 2^3 = 8 \quad \text{(here } \sqrt[2]{\ } = \sqrt{\ }\text{)}
  • x5/2=x5=(x)5if x≥0x^{5/2} = \sqrt{x^5} = (\sqrt{x})^5 \quad \text{if } x \geq 0
  • (19)1/2=19=13\left(\dfrac{1}{9}\right)^{1/2} = \sqrt{\dfrac{1}{9}} = \dfrac{1}{3}

Exponentiation

Fundamental Identity

P=an\boxed{P = a^n}

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

Where:

  • aa: base
  • nn: natural exponent
  • PP: power

Properties

Product of equal bases

am⋅an=am+n\boxed{a^m \cdot a^n = a^{m+n}}

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

📝 Examples

  • 23⋅24=23+4=27=1282^3 \cdot 2^4 = 2^{3+4} = 2^7 = 128
  • x5⋅x2=x5+2=x7x^5 \cdot x^2 = x^{5+2} = x^7
  • (−3)2⋅(−3)6=(−3)2+6=(−3)8=6561(-3)^2 \cdot (-3)^6 = (-3)^{2+6} = (-3)^8 = 6561
  • a4⋅a1=a4+1=a5a^4 \cdot a^1 = a^{4+1} = a^5
  • (12)3⋅(12)5=(12)3+5=(12)8=1256\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}
  • yn⋅y7=yn+7y^{n} \cdot y^{7} = y^{n+7}

Power of a power

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

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

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

📝 Examples

  • (23)4=23⋅4=212=4096(2^3)^4 = 2^{3 \cdot 4} = 2^{12} = 4096
  • (x2)5=x2⋅5=x10(x^2)^5 = x^{2 \cdot 5} = x^{10}
  • ((−3)4)2=(−3)4⋅2=(−3)8=6561((-3)^4)^2 = (-3)^{4 \cdot 2} = (-3)^8 = 6561
  • (a3)1=a3⋅1=a3(a^3)^1 = a^{3 \cdot 1} = a^3
  • ((12)2)3=(12)2⋅3=(12)6=164\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}
  • (ym)7=ym⋅7=y7m(y^m)^7 = y^{m \cdot 7} = y^{7m}

Power of a product

(a⋅b)n=an⋅bn\boxed{(a \cdot b)^n = a^n \cdot b^n}

a,b∈R,n∈Na, b \in \mathbb{R}, n \in \mathbb{N}

(xa⋅yb)n=xa⋅n⋅yb⋅n\boxed{(x^a \cdot y^b)^n = x^{a \cdot n} \cdot y^{b \cdot n}}

📝 Examples

  • (3⋅4)2=32⋅42=9⋅16=144(3 \cdot 4)^2 = 3^2 \cdot 4^2 = 9 \cdot 16 = 144
  • (2x)3=23⋅x3=8x3(2x)^3 = 2^3 \cdot x^3 = 8x^3
  • (−5⋅2)4=(−5)4⋅24=625⋅16=10000(-5 \cdot 2)^4 = (-5)^4 \cdot 2^4 = 625 \cdot 16 = 10000
  • (ab)5=a5⋅b5(ab)^5 = a^5 \cdot b^5
  • (12⋅y)3=(12)3⋅y3=18y3\left(\dfrac{1}{2} \cdot y\right)^3 = \left(\dfrac{1}{2}\right)^3 \cdot y^3 = \dfrac{1}{8}y^3
  • (xy2)4=x4⋅(y2)4=x4⋅y8(xy^2)^4 = x^4 \cdot (y^2)^4 = x^4 \cdot y^8

Division of equal bases

aman=am−n\boxed{\frac{a^m}{a^n} = a^{m-n}}

m,n∈N,m≥n,a∈R∖{0}m, n \in \mathbb{N}, m \geq n, a \in \mathbb{R} \setminus \{0\}

📝 Examples

  • 5753=57−3=54=625\dfrac{5^7}{5^3} = 5^{7-3} = 5^4 = 625
  • x8x5=x8−5=x3\dfrac{x^8}{x^5} = x^{8-5} = x^3
  • (−2)6(−2)2=(−2)6−2=(−2)4=16\dfrac{(-2)^6}{(-2)^2} = (-2)^{6-2} = (-2)^4 = 16
  • a10a10=a10−10=a0=1(if a≠0)\dfrac{a^{10}}{a^{10}} = a^{10-10} = a^0 = 1 \quad \text{(if } a \neq 0\text{)}
  • 3532=35−2=33=27\dfrac{3^5}{3^2} = 3^{5-2} = 3^3 = 27
  • yn+4yn=y(n+4)−n=y4\dfrac{y^{n+4}}{y^n} = y^{(n+4)-n} = y^4

Power of a quotient

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

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

📝 Examples

  • (23)4=2434=1681\left(\dfrac{2}{3}\right)^4 = \dfrac{2^4}{3^4} = \dfrac{16}{81}
  • (52)3=5323=1258\left(\dfrac{5}{2}\right)^3 = \dfrac{5^3}{2^3} = \dfrac{125}{8}
  • (x4)2=x242=x216if x∈R\left(\dfrac{x}{4}\right)^2 = \dfrac{x^2}{4^2} = \dfrac{x^2}{16} \quad \text{if } x \in \mathbb{R}
  • (−35)3=(−3)353=−27125\left(\dfrac{-3}{5}\right)^3 = \dfrac{(-3)^3}{5^3} = \dfrac{-27}{125}
  • (ab)5=a5b5if b≠0\left(\dfrac{a}{b}\right)^5 = \dfrac{a^5}{b^5} \quad \text{if } b \neq 0
  • (1x)n=1nxn=1xnif x≠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

(ab)−n=(ba)n=bnan\boxed{\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n = \frac{b^n}{a^n}}

a,b≠0a, b \neq 0

📝 Examples

  • (23)−4=(32)4=3424=8116\left(\dfrac{2}{3}\right)^{-4} = \left(\dfrac{3}{2}\right)^4 = \dfrac{3^4}{2^4} = \dfrac{81}{16}
  • (5x)−2=(x5)2=x225if x≠0\left(\dfrac{5}{x}\right)^{-2} = \left(\dfrac{x}{5}\right)^2 = \dfrac{x^2}{25} \quad \text{if } x \neq 0
  • (14)−3=(41)3=43=64\left(\dfrac{1}{4}\right)^{-3} = \left(\dfrac{4}{1}\right)^3 = 4^3 = 64
  • (ab)−1=baif a,b≠0\left(\dfrac{a}{b}\right)^{-1} = \dfrac{b}{a} \quad \text{if } a, b \neq 0
  • (−27)−3=(7−2)3=73(−2)3=343−8=−3438\left(\dfrac{-2}{7}\right)^{-3} = \left(\dfrac{7}{-2}\right)^3 = \dfrac{7^3}{(-2)^3} = \dfrac{343}{-8} = -\dfrac{343}{8}
  • (xy)−5=y5x5if x,y≠0\left(\dfrac{x}{y}\right)^{-5} = \dfrac{y^5}{x^5} \quad \text{if } x, y \neq 0

Successive Exponents

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

📝 Examples

  • 2231=223=28=2562^{2^{3^1}} = 2^{2^3} = 2^8 = 256
  • 3222=324=316=43, ⁣046, ⁣7213^{2^{2^2}} = 3^{2^4} = 3^{16} = 43,\!046,\!721
  • 5142=5116=51=55^{1^{4^2}} = 5^{1^{16}} = 5^1 = 5
  • 10315=1031=103=1, ⁣00010^{3^{1^5}} = 10^{3^1} = 10^3 = 1,\!000
  • 2321=232=29=5122^{3^{2^1}} = 2^{3^2} = 2^9 = 512
  • 4230=421=42=164^{2^{3^0}} = 4^{2^1} = 4^2 = 16

Absolute Value

x2k2k=∣x∣={xif: x>00if: x=0−xif: x<0\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}}

🧠

Remember

Commonly used powers
22=42^2 = 4 32=93^2 = 9 42=164^2 = 16 82=648^2 = 64
23=82^3 = 8 33=273^3 = 27 43=644^3 = 64 83=5128^3 = 512
24=162^4 = 16 34=813^4 = 81 52=255^2 = 25 92=819^2 = 81
25=322^5 = 32 35=2433^5 = 243 53=1255^3 = 125 93=7299^3 = 729
26=642^6 = 64 36=7293^6 = 729 54=6255^4 = 625 102=10010^2 = 100
27=1282^7 = 128 55=31255^5 = 3125 103=100010^3 = 1000
28=2562^8 = 256 62=366^2 = 36 112=12111^2 = 121
29=5122^9 = 512 63=2166^3 = 216 113=133111^3 = 1331
210=10242^{10} = 1024 64=12966^4 = 1296 122=14412^2 = 144
72=497^2 = 49 123=172812^3 = 1728
73=3437^3 = 343
74=24017^4 = 2401

Radicals in R\mathbb{R}

Fundamental Identity

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

Where:

  • xx: radicand (x∈R,x≥0x \in \mathbb{R}, x \geq 0 if nn is even)
  • nn: index (n∈N,n≥2n \in \mathbb{N}, n \geq 2)
  • yy: root (y≥0y \geq 0 if nn is even)
🧠

Remember

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

Examples:

  • 0=0\sqrt{0} = 0
  • 03=0\sqrt[3]{0} = 0
  • 05=0\sqrt[5]{0} = 0
  • 07=0\sqrt[7]{0} = 0

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

Examples:

  • 1=1\sqrt{1} = 1
  • 13=1\sqrt[3]{1} = 1
  • 15=1\sqrt[5]{1} = 1
  • 17=1\sqrt[7]{1} = 1

Sign rule:

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

(−)2n+1=(−)\sqrt[2n+1]{(-)} = (-)

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

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

In summary:

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

−odd=−\sqrt[\text{odd}]{-} = -

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

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

Properties

Root of a product

a⋅bn=an⋅bn\boxed{\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}}

If nn is even, then a≥0a \geq 0 and b≥0b \geq 0.

📝 Examples

  • 4⋅9=4⋅9=2⋅3=6\sqrt{4 \cdot 9} = \sqrt{4} \cdot \sqrt{9} = 2 \cdot 3 = 6
  • 8⋅273=83⋅273=2⋅3=6\sqrt[3]{8 \cdot 27} = \sqrt[3]{8} \cdot \sqrt[3]{27} = 2 \cdot 3 = 6
  • 16⋅814=164⋅814=2⋅3=6\sqrt[4]{16 \cdot 81} = \sqrt[4]{16} \cdot \sqrt[4]{81} = 2 \cdot 3 = 6
  • x⋅y=x⋅yif x≥0, y≥0\sqrt{x \cdot y} = \sqrt{x} \cdot \sqrt{y} \quad \text{if } x \geq 0,\ y \geq 0
  • a5⋅b105=a55⋅b105=a⋅b2\sqrt[5]{a^5 \cdot b^{10}} = \sqrt[5]{a^5} \cdot \sqrt[5]{b^{10}} = a \cdot b^2
  • −8⋅643=−83⋅643=(−2)⋅4=−8\sqrt[3]{-8 \cdot 64} = \sqrt[3]{-8} \cdot \sqrt[3]{64} = (-2) \cdot 4 = -8
🧠

Remember

Square Roots (simplified forms)

  • 22=82\sqrt{2} =\sqrt{8}
  • 32=183\sqrt{2} =\sqrt{18}
  • 42=324\sqrt{2} =\sqrt{32}
  • 52=505\sqrt{2} =\sqrt{50}
  • 33=273\sqrt{3} =\sqrt{27}
  • 43=484\sqrt{3} =\sqrt{48}
  • 53=755\sqrt{3} =\sqrt{75}
  • 35=453\sqrt{5} =\sqrt{45}
  • 45=804\sqrt{5} =\sqrt{80}
  • 25=202\sqrt{5} =\sqrt{20}

Cube Roots (simplified forms)

  • 223=1632\sqrt[3]{2} =\sqrt[3]{16}
  • 323=5433\sqrt[3]{2} =\sqrt[3]{54}
  • 233=2432\sqrt[3]{3} =\sqrt[3]{24}
  • 333=8133\sqrt[3]{3} =\sqrt[3]{81}

Approximate decimal values (square roots)

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

Root of a quotient

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

b≠0b \neq 0

If nn is even, then a≥0a \geq 0 and b>0b > 0.

📝 Examples

  • 916=916=34\sqrt{\dfrac{9}{16}} = \dfrac{\sqrt{9}}{\sqrt{16}} = \dfrac{3}{4}
  • 271253=2731253=35\sqrt[3]{\dfrac{27}{125}} = \dfrac{\sqrt[3]{27}}{\sqrt[3]{125}} = \dfrac{3}{5}
  • 1814=14814=13\sqrt[4]{\dfrac{1}{81}} = \dfrac{\sqrt[4]{1}}{\sqrt[4]{81}} = \dfrac{1}{3}
  • x5325=x55325=x2if x∈R\sqrt[5]{\dfrac{x^5}{32}} = \dfrac{\sqrt[5]{x^5}}{\sqrt[5]{32}} = \dfrac{x}{2} \quad \text{if } x \in \mathbb{R}
  • ab=abif a≥0, b>0\sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}} \quad \text{if } a \geq 0,\ b > 0
  • −8273=−83273=−23\sqrt[3]{\dfrac{-8}{27}} = \dfrac{\sqrt[3]{-8}}{\sqrt[3]{27}} = \dfrac{-2}{3}

Root of a root

anm=am⋅n\boxed{\sqrt[m]{\sqrt[n]{a}} = \sqrt[m \cdot n]{a}}

m,n∈Nm, n \in \mathbb{N}

If m⋅nm \cdot n is even, then a≥0a \geq 0.

arsnm=am⋅n⋅s⋅r\boxed{\sqrt[m]{\sqrt[n]{\sqrt[s]{\sqrt[r]{a}}}} = \sqrt[m \cdot n \cdot s \cdot r]{a}}

📝 Examples

  • 832=86=236=23/6=21/2=2\sqrt[2]{\sqrt[3]{8}} = \sqrt[6]{8} = \sqrt[6]{2^3} = 2^{3/6} = 2^{1/2} = \sqrt{2}
  • x43=x12for x≥0\sqrt[3]{\sqrt[4]{x}} = \sqrt[12]{x} \quad \text{for } x \geq 0
  • 16=1622=164=2\sqrt{\sqrt{16}} = \sqrt[2]{\sqrt[2]{16}} = \sqrt[4]{16} = 2
  • a5=a10if a≥0\sqrt[5]{\sqrt{a}} = \sqrt[10]{a} \quad \text{if } a \geq 0
  • 6434=6412=2612=26/12=21/2=2\sqrt[4]{\sqrt[3]{64}} = \sqrt[12]{64} = \sqrt[12]{2^6} = 2^{6/12} = 2^{1/2} = \sqrt{2}
  • x=x222=x8for x≥0\sqrt{\sqrt{\sqrt{x}}} = \sqrt[2]{\sqrt[2]{\sqrt[2]{x}}} = \sqrt[8]{x} \quad \text{for } x \geq 0

Power of a root

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

xacab=xcbIf ab is even, then x∈R0+\boxed{\sqrt[ab]{x^{ac}} = \sqrt[b]{x^c} \quad \text{If } ab \text{ is even, then } x \in \mathbb{R}_0^+}

📝 Examples

  • (23)6=263=643=4\left(\sqrt[3]{2}\right)^6 = \sqrt[3]{2^6} = \sqrt[3]{64} = 4
  • (5)4=54=625=25\left(\sqrt{5}\right)^4 = \sqrt{5^4} = \sqrt{625} = 25
  • (34)2=324=94\left(\sqrt[4]{3}\right)^2 = \sqrt[4]{3^2} = \sqrt[4]{9}
  • (xn)3=x3nif x≥0 when n is even\left(\sqrt[n]{x}\right)^3 = \sqrt[n]{x^3} \quad \text{if } x \geq 0 \text{ when } n \text{ is even}
  • (−325)3=(−32)35=−327685=−8\left(\sqrt[5]{-32}\right)^3 = \sqrt[5]{(-32)^3} = \sqrt[5]{-32768} = -8
  • (a2+b2)2=(a2+b2)2=∣a2+b2∣=a2+b2\left(\sqrt{a^2 + b^2}\right)^2 = \sqrt{(a^2 + b^2)^2} = |a^2 + b^2| = a^2 + b^2

Successive Radicals

xa⋅xb⋅xcpmn=xa⋅m⋅p+b⋅p+cnmp\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}}

xa÷xb÷xcpmn=xa⋅m⋅p−b⋅p+cnmp\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}}

x⋅y⋅zpnm=xm⋅ym⋅n⋅zm⋅n⋅p\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}

xa⋅yb⋅zcpnm=xam⋅ybm⋅n⋅zcm⋅n⋅p\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}

xa÷yb÷zc÷wd÷ve÷ufsrqpnm=xam⋅zcmnp⋅vemnpqrybmn⋅wdmnpq⋅ufmnpqrs\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

xm⋅ypn=xmn⋅ypnx^m \cdot \sqrt[n]{y^p} = \sqrt[n]{x^{mn} \cdot y^p}

xm⋅xn=xn+mmn\sqrt[m]{x} \cdot \sqrt[n]{x} = \sqrt[mn]{x^{n+m}}

xmxn=xn−mmn, x≠0\frac{\sqrt[m]{x}}{\sqrt[n]{x}} = \sqrt[mn]{x^{n-m}}, \, x \neq 0

1zcybxa=1x⋅1y⋅1zcba\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}}}}

a⋅b⋅cpnmd⋅e⋅fpnm=ad⋅be⋅cfpnm\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}}}}

xyzpnm=xmynzp\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

…xnnnn⏟m radicals=xnm\underbrace{\sqrt[n]{ \sqrt[n]{ \sqrt[n]{ \dots \sqrt[n]{x}}}}}_{\text{m radicals}} = \sqrt[n^m]{x}

x⋅x⋅x⋅...⋅xnnnn⏟m radicals=xnm−1n−1nm\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}}}

x÷x÷x÷...⋅÷xnnnn⏟′′m′′ radicals={xnm−1n+1nmIf ’m’ is even,xnm+1n+1nmIf ’m’ is odd.\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}

  1. Expressions with infinite radicals

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

x⋅(x+1)−x⋅(x+1)−x⋅(x+1)−…=x\sqrt{x\cdot(x+1)- \sqrt{x\cdot(x+1)- \sqrt{x\cdot(x+1)- \dots } }} = x

x⋅x⋅x⋅...nnn=xn−1\sqrt[n]{x\cdot \sqrt[n]{x\cdot \sqrt[n]{x\cdot...} }} = \sqrt[n-1]{x}

x÷x÷x÷...nnn=xn+1\sqrt[n]{x\div \sqrt[n]{x\div \sqrt[n]{x\div...} }} = \sqrt[n+1]{x}

a⋅a⋅a⋅a⋅…mnmn=am+1m⋅n−1\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}}

a⋅b⋅a⋅b⋅…mnmn=am⋅bm⋅n−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}

a+a+a+…=1+4a+12\quad \sqrt{a + \sqrt{a + \sqrt{a + \dots}}} = \frac{1 + \sqrt{4a + 1}}{2}

a−a−a−…=−1+4a+12\quad \sqrt{a - \sqrt{a - \sqrt{a - \dots}}} = \frac{-1 + \sqrt{4a + 1}}{2}

  1. Continuous infinite exponential

xxx...xn=n→x=nn; x≠0x^{x^{x^{{.}^{{.}^{{.}^{x^n}}}}}} = n \quad \rightarrow \quad x = \sqrt[n]{n}; \, x \neq 0

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