Algebra Equation Sheet
Exponent Properties
\[
a^na^m=a^{n+m}
\]
\[
\frac{a^n}{a^m}=a^{n-m}
\]
\[
(a^n)^m=a^{nm}
\]
\[
a^0=1
\]
\[
(ab)^n=a^nb^n
\]
\[
\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}
\]
\[
a^{-1}=\frac{1}{a^n}
\]
Log Properties
\[
\log_b(x)=y\Leftrightarrow x=b^y
\]
\[
\ln=\log_e
\]
\[
\log=\log_{10}
\]
\[
\log_b(b)=1
\]
\[
\log_b{1}=0
\]
\[
\log_b(b^x)=x
\]
\[
b^{\log_b(x)}=x
\]
\[
\log_b(x^a)=a\log_b(x)
\]
\[
\log_b(xy)=\log_b(x)+\log_b(y)
\]
\[
\log_b\left(\frac{x}{y}\right)=\log_b(x)-\log_b(y)
\]
Factoring
\[
x^2+2ax+a^2=(x+a)^2
\]
\[
x^2-2ax+a^2=(x-a)^2
\]
\[
x^2-a^2=(x+a)(x-a)
\]
\[
x^2+(a+b)x+ab=(x+a)(x+b)
\]
\[
x^3+3ax^2+3a^2x+a^3=(x+a)^3
\]
\[
x^3-3ax^2+3a^2x-a^3=(x-a)^3
\]
\[
(x^2-ax+a^2)(x+a)=x^3+a^3
\]
\[
(x^2+ax+a^2)(x-a)=x^3-a^3
\]
Rad Properties
\[
\sqrt[n]{a}=a^{1/n}
\]
\[
\sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}
\]
\[
\sqrt[m]{\sqrt[n]{a}}=\sqrt[nm]{a}
\]
Quadratic Formula
To Solve \( ax^2+bx+c=0 \)
\[
x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}
\]