Algebra

Algebra Formulas

The most used algebra rules in one place. Details and solved problems are in the lesson for each group.

Removing brackets

Rule	Formula
Minus before minus	$-(-a)=a$
Minus before brackets	${-(a+b)}={-a-b},$ ${-(a-b)}={-a+b}$
Multiplying a sum	${a(b+c)}={ab+ac}$
Multiplying two sums	${(a+b)(c+d)}={ac+ad+bc+bd}$

Special products and factoring

Rule	Formula
Square of a sum	${(a+b)^2}={a^2+2ab+b^2}$
Square of a difference	${(a-b)^2}={a^2-2ab+b^2}$
Difference of squares	${a^2-b^2}={(a-b)(a+b)}$
Cube of a sum	${(a+b)^3}={a^3+3a^2b+3ab^2+b^3}$
Cube of a difference	${(a-b)^3}={a^3-3a^2b+3ab^2-b^3}$
Sum of cubes	${a^3+b^3}={(a+b)(a^2-ab+b^2)}$
Difference of cubes	${a^3-b^3}={(a-b)(a^2+ab+b^2)}$

The sum of squares $a^2+b^2$ cannot be factored (with real numbers). Do not confuse: ${(a+b)^2\ne a^2+b^2}.$ More in the lesson Polynomial identities.

Fractions

All denominators are nonzero.

Rule	Formula
Expanding and reducing	${\frac{a}{b}}={\frac{a\cdot k}{b\cdot k}},$ $k\ne 0$
Addition and subtraction	${\frac{a}{b}\pm\frac{c}{d}}={\frac{ad\pm bc}{bd}}$
Multiplication	${\frac{a}{b}\cdot\frac{c}{d}}={\frac{ac}{bd}}$
Division	${\frac{a}{b}\div\frac{c}{d}}={\frac{a}{b}\cdot\frac{d}{c}}={\frac{ad}{bc}},$ $c\ne 0$
Sign of a fraction	${\frac{-a}{b}}={\frac{a}{-b}}={-\frac{a}{b}},$ ${\frac{-a}{-b}}={\frac{a}{b}}$
Zero numerator	${\frac{0}{b}}=0$
Zero denominator	$\frac{a}{0}$ is undefined: you cannot divide by 0
Complex fraction	${\dfrac{\frac{a}{b}}{\frac{c}{d}}}={\dfrac{ad}{bc}},$ ${\dfrac{a}{\frac{b}{c}}}={\dfrac{ac}{b}}$

Exponents

Here $a\ne 0,$ $b\ne 0,$ and $m$ and $n$ are integers.

Rule	Formula
Product of powers with the same base	${a^m\cdot a^n}={a^{m+n}}$
Quotient of powers with the same base	${a^m\div a^n}={a^{m-n}}$
Power of a power	${(a^m)^n}={a^{mn}}$
Power of a product	${(ab)^n}={a^nb^n}$
Power of a quotient	${\left(\frac{a}{b}\right)^n}={\frac{a^n}{b^n}}$
Zero exponent	${a^0}=1$
Negative exponent	${a^{-n}}={\frac{1}{a^n}},$ ${\left(\frac{a}{b}\right)^{-n}}={\left(\frac{b}{a}\right)^{n}}$
Base 1 and base 0	${1^n}=1;$ ${0^n}=0$ for $n\gt 0,$ while $0^0$ and $0^{-n}$ are undefined

Roots

For even $n$ the numbers under the root must be nonnegative: the square root of a negative number is not a real number.

Rule	Formula
Definition	${\left(\sqrt[n]{a}\right)^n}=a$
Root of a power	${\sqrt{a^2}}={\|a\|},$ ${\sqrt[n]{a^n}}=a$ for $a\ge 0$
Root of a product	${\sqrt[n]{ab}}={\sqrt[n]{a}\cdot\sqrt[n]{b}}$
Root of a quotient	${\sqrt[n]{\frac{a}{b}}}={\frac{\sqrt[n]{a}}{\sqrt[n]{b}}},$ $b\ne 0$
Root of a root	${\sqrt[m]{\sqrt[n]{a}}}={\sqrt[mn]{a}}$
Fractional exponent	${a^{\frac{m}{n}}}={\sqrt[n]{a^m}}={\left(\sqrt[n]{a}\right)^m},$ $a\gt 0$

Absolute value

Rule	Formula
Definition	$\|x\|=x$ if $x\ge 0;$ $\|x\|=-x$ if $x\lt 0$
Opposite numbers	${\|-a\|}={\|a\|}$
Product and quotient	${\|ab\|}={\|a\|\cdot\|b\|},$ ${\left\|\frac{a}{b}\right\|}={\frac{\|a\|}{\|b\|}},$ $b\ne 0$
Equation and inequality	${\|x\|=a}\iff{x=\pm a}$ for $a\ge 0;$ ${\|x\|\lt a}\iff{-a\lt x\lt a}$

Logarithms

Here $a\gt 0,$ $a\ne 1,$ $x\gt 0$ and $y\gt 0.$

Rule	Formula
Definition	${\log_a x=b}\iff{a^b=x}$
Logarithm of 1 and of the base	${\log_a 1}=0,$ ${\log_a a}=1$
Logarithm of a product	${\log_a(xy)}={\log_a x+\log_a y}$
Logarithm of a quotient	${\log_a\frac{x}{y}}={\log_a x-\log_a y}$
Logarithm of a power	${\log_a x^n}={n\log_a x}$
Change of base	${\log_a x}={\frac{\log_b x}{\log_b a}},$ $b\gt 0,$ $b\ne 1$

Factorial

Rule	Formula
Definition	${n!}={1\cdot 2\cdot 3\cdots n}$ for $n\ge 1$
Zero factorial	${0!}=1$
Recurrence	${(n+1)!}={(n+1)\cdot n!}$

Practice

Interactive exercises with answer checking: