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Algebra

Algebra Formulas

Introduction to Algebra Section

Algebra is a cornerstone of mathematics that explores relationships between quantities and provides tools to solve problems logically and systematically. It begins with basic operations like addition, subtraction, multiplication, and division but quickly extends to include powers, roots, and logarithms. Central to algebra is the use of equations and inequalities to model and solve problems, helping us determine unknown values based on given conditions.

The study of algebra introduces polynomials, expressions composed of variables and constants combined through arithmetic operations. Techniques like factoring and expanding polynomials allow us to simplify and solve complex equations. Algebra also explores systems of equations, where multiple relationships are analyzed simultaneously to find solutions that satisfy all constraints.

Functions are another key concept in algebra, providing a way to describe how quantities depend on each other. Linear functions, quadratic functions, and exponential relationships reveal patterns and behaviors that are essential for deeper mathematical understanding. Algebra also emphasizes the properties of numbers and operations, such as commutativity and distributivity, which underpin all calculations.

The skills developed in algebra, such as logical reasoning, abstraction, and problem-solving, are invaluable. They find applications in diverse fields, from science and engineering to economics and data analysis, forming a crucial foundation for advanced mathematical studies.

Algebra Formulas

Explore Algebra formulas with explanations and examples

Product Rule

xm⋅xn=xm+nx^m \cdot x^n = x^{m+n}

Quotient Rule

xmxn=xm−n\frac{x^m}{x^n} = x^{m-n}

Power Rule

(xm)n=xm⋅n(x^m)^n = x^{m \cdot n}

Zero Exponent Rule

x0=1,  x∈R∖{0}x^0 = 1, \; x \in \mathbb{R} \setminus \{0\}

Negative Exponent Rule

x−n=1xnx^{-n} = \frac{1}{x^n}

Fractional Exponent Rule

xmn=xmnx^{\frac{m}{n}} = \sqrt[n]{x^m}

Product to Power Rule

(xy)n=xnyn(xy)^n = x^n y^n

Product Rule for Radicals

xyn=xn⋅yn\sqrt[n]{xy} = \sqrt[n]{x} \cdot \sqrt[n]{y}

Quotient Rule for Radicals

xyn=xnyn\sqrt[n]{\frac{x}{y}} = \frac{\sqrt[n]{x}}{\sqrt[n]{y}}

Power Rule for Radicals

xmn=xmn\sqrt[n]{x^m} = x^{\frac{m}{n}}

Root of a Root Rule

xnm=xmn\sqrt[m]{\sqrt[n]{x}} = \sqrt[mn]{x}

Like Root Addition Rule

axn+bxn=(a+b)xna\sqrt[n]{x} + b\sqrt[n]{x} = (a+b)\sqrt[n]{x}

Even/Odd Root Property

(−x)n={−xnif n is oddundefined over Rif n is even and x > 0\sqrt[n]{(-x)} = \begin{cases} -\sqrt[n]{x} & \text{if n is odd} \\ \text{undefined over } \mathbb{R} & \text{if n is even and x > 0} \end{cases}

Rationalization Rule

abn=abn−1nbnn=abn−1nb\frac{a}{\sqrt[n]{b}} = \frac{a\sqrt[n]{b^{n-1}}}{\sqrt[n]{b^n}} = \frac{a\sqrt[n]{b^{n-1}}}{b}

Basic Definition of logarithm

y=log⁡bx  ⟺  by=xy = \log_b x \iff b^y = x

Product Rule for Logarithms

log⁡b(MN)=log⁡bM+log⁡bN\log_b(MN) = \log_b M + \log_b N

Quotient Rule for Logarithms

log⁡b(MN)=log⁡bM−log⁡bN\log_b(\frac{M}{N}) = \log_b M - \log_b N

Power Rule for Logarithms

log⁡b(Mp)=plog⁡bM\log_b(M^p) = p\log_b M

Change of Base

log⁡bM=log⁡kMlog⁡kb\log_b M = \frac{\log_k M}{\log_k b}

Special Values

log⁡bb=1,log⁡b1=0\log_b b = 1,\quad \log_b 1 = 0

Binomial Theorem

(x+y)n=∑k=0n(nk)xn−kyk=(n0)xn+(n1)xn−1y+(n2)xn−2y2+...+(nn)yn(x + y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k}y^k = \binom{n}{0}x^n + \binom{n}{1}x^{n-1}y + \binom{n}{2}x^{n-2}y^2 + ... + \binom{n}{n}y^n

Binomial Coefficient Formula

(nk)=n!k!(n−k)!,  n,k∈N0,  k≤n\binom{n}{k} = \frac{n!}{k!(n-k)!}, \; n,k \in \mathbb{N}_0, \; k \leq n

Square of Binomial

(x+y)2=x2+2xy+y2(x + y)^2 = x^2 + 2xy + y^2

Square of Difference

(x−y)2=x2−2xy+y2(x - y)^2 = x^2 - 2xy + y^2

Product of Sum and Difference

(x+y)(x−y)=x2−y2(x + y)(x - y) = x^2 - y^2

Cube of Binomial

(x+y)3=x3+3x2y+3xy2+y3(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3

Cube of Difference

(x−y)3=x3−3x2y+3xy2−y3(x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3

Product Rule

xm⋅xn=xm+nx^m \cdot x^n = x^{m+n}

Quotient Rule

xmxn=xm−n\frac{x^m}{x^n} = x^{m-n}

Power Rule

(xm)n=xm⋅n(x^m)^n = x^{m \cdot n}

Zero Exponent Rule

x0=1,  x∈R∖{0}x^0 = 1, \; x \in \mathbb{R} \setminus \{0\}

Negative Exponent Rule

x−n=1xnx^{-n} = \frac{1}{x^n}

Fractional Exponent Rule

xmn=xmnx^{\frac{m}{n}} = \sqrt[n]{x^m}

Product to Power Rule

(xy)n=xnyn(xy)^n = x^n y^n

Product Rule for Radicals

xyn=xn⋅yn\sqrt[n]{xy} = \sqrt[n]{x} \cdot \sqrt[n]{y}

Quotient Rule for Radicals

xyn=xnyn\sqrt[n]{\frac{x}{y}} = \frac{\sqrt[n]{x}}{\sqrt[n]{y}}

Power Rule for Radicals

xmn=xmn\sqrt[n]{x^m} = x^{\frac{m}{n}}

Root of a Root Rule

xnm=xmn\sqrt[m]{\sqrt[n]{x}} = \sqrt[mn]{x}

Like Root Addition Rule

axn+bxn=(a+b)xna\sqrt[n]{x} + b\sqrt[n]{x} = (a+b)\sqrt[n]{x}

Even/Odd Root Property

(−x)n={−xnif n is oddundefined over Rif n is even and x > 0\sqrt[n]{(-x)} = \begin{cases} -\sqrt[n]{x} & \text{if n is odd} \\ \text{undefined over } \mathbb{R} & \text{if n is even and x > 0} \end{cases}

Rationalization Rule

abn=abn−1nbnn=abn−1nb\frac{a}{\sqrt[n]{b}} = \frac{a\sqrt[n]{b^{n-1}}}{\sqrt[n]{b^n}} = \frac{a\sqrt[n]{b^{n-1}}}{b}

Basic Definition of logarithm

y=log⁡bx  ⟺  by=xy = \log_b x \iff b^y = x

Product Rule for Logarithms

log⁡b(MN)=log⁡bM+log⁡bN\log_b(MN) = \log_b M + \log_b N

Quotient Rule for Logarithms

log⁡b(MN)=log⁡bM−log⁡bN\log_b(\frac{M}{N}) = \log_b M - \log_b N

Power Rule for Logarithms

log⁡b(Mp)=plog⁡bM\log_b(M^p) = p\log_b M

Change of Base

log⁡bM=log⁡kMlog⁡kb\log_b M = \frac{\log_k M}{\log_k b}

Special Values

log⁡bb=1,log⁡b1=0\log_b b = 1,\quad \log_b 1 = 0

Binomial Theorem

(x+y)n=∑k=0n(nk)xn−kyk=(n0)xn+(n1)xn−1y+(n2)xn−2y2+...+(nn)yn(x + y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k}y^k = \binom{n}{0}x^n + \binom{n}{1}x^{n-1}y + \binom{n}{2}x^{n-2}y^2 + ... + \binom{n}{n}y^n

Binomial Coefficient Formula

(nk)=n!k!(n−k)!,  n,k∈N0,  k≤n\binom{n}{k} = \frac{n!}{k!(n-k)!}, \; n,k \in \mathbb{N}_0, \; k \leq n

Square of Binomial

(x+y)2=x2+2xy+y2(x + y)^2 = x^2 + 2xy + y^2

Square of Difference

(x−y)2=x2−2xy+y2(x - y)^2 = x^2 - 2xy + y^2

Product of Sum and Difference

(x+y)(x−y)=x2−y2(x + y)(x - y) = x^2 - y^2

Cube of Binomial

(x+y)3=x3+3x2y+3xy2+y3(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3

Cube of Difference

(x−y)3=x3−3x2y+3xy2−y3(x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3