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Jumampi

Exponent Theory

Natural Exponent

Section titled “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} }

Zero Exponent

Section titled “Zero Exponent”
a0=1∀a∈R∖{0}a^0 = 1 \quad \forall a \in \mathbb{R} \setminus \{0\}

Negative Exponent

Section titled “Negative Exponent”
a−n=1an∀a∈R∖{0},n∈Na^{-n} = \frac{1}{a^n} \quad \forall a \in \mathbb{R} \setminus \{0\}, n \in \mathbb{N}

Fractional Exponent

Section titled “Fractional Exponent”
am/n=amn=(an)m∀n∈N,n≥2a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m \quad \forall n \in \mathbb{N}, n \geq 2

Exponentiation

Section titled “Exponentiation”

Fundamental Identity

Section titled “Fundamental Identity”
P=anwhere: a∈R,n∈N,P∈RP = a^n \quad \text{where: } a \in \mathbb{R}, n \in \mathbb{N}, P \in \mathbb{R}
  • aa: base
  • nn: natural exponent
  • PP: power

Properties

Section titled “Properties”
  1. Product of equal bases:
xm⋅xn=xm+nx∈R,m,n∈Nx^m \cdot x^n = x^{m+n} \quad x \in \mathbb{R}, m, n \in \mathbb{N}
  1. Exponent of an exponent:
(xm)n=xm⋅nx∈R,m,n∈N(x^m)^n = x^{m \cdot n} \quad x \in \mathbb{R}, m, n \in \mathbb{N} (((am)n)r)s=am⋅n⋅r⋅s\left(((a^m)^n)^r\right)^s = a^{m \cdot n \cdot r \cdot s}
  1. Exponent of a product:
(a⋅b)n=an⋅bna,b∈R,n∈N(a \cdot b)^n = a^n \cdot b^n \quad a, b \in \mathbb{R}, n \in \mathbb{N} (xa⋅yb)n=xa⋅n⋅yb⋅n(x^a \cdot y^b)^n = x^{a \cdot n} \cdot y^{b \cdot n}
  1. Division of equal bases:
aman=am−nm,n∈N,m≥n,a∈R∖{0}\frac{a^m}{a^n} = a^{m-n} \quad m, n \in \mathbb{N}, m \geq n, a \in \mathbb{R} \setminus \{0\}
  1. Exponent of a quotient:
(ab)n=anbnn∈N,b∈R∖{0}\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \quad n \in \mathbb{N}, b \in \mathbb{R} \setminus \{0\}
  1. Negative exponent of a fraction:
(ab)−n=(ba)n=bnana,b≠0\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n = \frac{b^n}{a^n} \quad a, b \neq 0

Radicals in R\mathbb{R}

Section titled “Radicals in R\mathbb{R}R”

Fundamental Identity

Section titled “Fundamental Identity”
y=xn  ⟺  yn=xn∈N,n≥2y = \sqrt[n]{x} \iff y^n = x \quad n \in \mathbb{N}, n \geq 2

Properties

Section titled “Properties”
  1. Root of a product:
a⋅bn=an⋅bn\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.

  1. Root of a quotient:
abn=anbnb≠0\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \quad b \neq 0

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

  1. Root of a root:
anm=am⋅nm,n∈N\sqrt[m]{\sqrt[n]{a}} = \sqrt[m \cdot n]{a} \quad m, n \in \mathbb{N}

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

arsnm=am⋅n⋅s⋅r\sqrt[m]{\sqrt[n]{\sqrt[s]{\sqrt[r]{a}}}} = \sqrt[m \cdot n \cdot s \cdot r]{a}
  1. Root of a power:
(an)m=amn\left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m} xacab=xcbIf ab is even, then x∈R0+\sqrt[ab]{x^{ac}} = \sqrt[b]{x^c} \quad \text{If } ab \text{ is even, then } x \in \mathbb{R}_0^+

Nested Radicals

Section titled “Nested Radicals”
a⋅b⋅cpmn=an⋅bn⋅m⋅cn⋅m⋅p\sqrt[n]{a \cdot \sqrt[m]{b \cdot \sqrt[p]{c}}} = \sqrt[n]{a} \cdot \sqrt[n \cdot m]{b} \cdot \sqrt[n \cdot m \cdot p]{c}

Exponential Equations

Section titled “Exponential Equations”
  1. Equality of bases:
am=an  ⟹  m=n(a≠0,1)a^m = a^n \implies m = n \quad (a \neq 0, 1)
  1. Equality of exponents:
am=bm  ⟹  a=b(m≠0)a^m = b^m \implies a = b \quad (m \neq 0)
  1. Special case:
ax=xa  ⟹  a=x(x≠0)a^x = x^a \implies a = x \quad (x \neq 0)

Equations with Variables in the Exponent

Section titled “Equations with Variables in the Exponent”
  1. Exponential form:
xx=an  ⟹  x=ax^x = a^n \implies x = a
  1. Radical form:
xx=n  ⟹  x=nxx^x = n \implies x = \sqrt[x]{n}