Question

test

Answer 1

Arithmetic operations
\[ab+ac=a(b+c)\]
\[a(\frac{ b }{ c })=\frac{ ab }{ c }\]
\[\frac{ \frac{ a }{ b } }{ \frac{ c }{ a } }=\frac{ a }{ bc }\]
\[\frac{ a }{ b }-\frac{ c }{ d }=\frac{ ab-bc }{ bd }\]
\[\frac{ a-b }{ c-d }=\frac{ b-a }{ d-c }\]
\[\frac{ a+b }{ c }=\frac{ a }{ c }+\frac{ b }{ c }\]
\[\frac{ ab+ac }{ a }=b+c; a \neq0\]
\[\frac{ a }{ \frac{ b }{ c } }=\frac{ ac }{ b }\]
\[\frac{ a }{ b }+\frac{ c }{ d }=\frac{ ad+bc }{ bd }\]

Quadratic Equation
\[ax^2+bx+c=0\]
\[x=\frac{ -b \pm \sqrt{b^2-4ac} }{ 2a }\]

Radicals
\[\sqrt[n]{a}=a^\frac{ 1 }{ n }\]
\[\sqrt[m]{\sqrt[n]{a}}=\sqrt[mn]{a}\]
\[\sqrt[n]{ab}=\sqrt[n]{a}*\sqrt[n]{b}\]
\[\sqrt[n]{\frac{ a }{ b }}=\frac{ \sqrt[n]{a} }{ \sqrt[n]{b} }\]
\[\sqrt[n]{a^{n}}=a; n=odd\]
\[\sqrt[n]{a^{n}}=|a|; n=even\]

\[a ^{n}a ^{m}=a ^{n+m}\]
\[(a ^{n})^{m}=a^{nm}\]
\[a^{-n}=\frac{ 1 }{ a ^{n} }\]
\[a^{0}=1\]
\[(\frac{ a }{ b })^{-n}=(\frac{ b }{ a })^{n}\]
\[\frac{ a^{n} }{ a^{m} }=a^{n-m}\]
\[\frac{ 1 }{ a ^{-n} }=a^{n}\]

Answer 2

\[i=\sqrt{-1}\]
\[i^{2}=\sqrt{-1}\]
\[\sqrt{-a}=i \sqrt{a}\]
\[(a+bi)-(c+di)=a-c+(b-d)i\]
\[(a+bi)+(c+di)=a+c+(b+d)i\]
\[(a+bi)(c+di)=ac-bd+(ad+bc)i\]
\[(a+bi)(c+di)=ac-bd+(ad+bc)i\]
\[(a+bi)(a-bi)=a^{2}+b^{2}\]
\[|a+bi|=\sqrt{a^{2}+b^{2}}\]
\[(a+bi)(a+bi)=|a+bi|^{2}\]
\[\frac{ 1 }{ (a+bi) }=\frac{ a-bi }{ a^{2}+b^{2} }\]

Answer 3

Logarithms
\[y=\log_{b}x; b^{y}=x\]
\[\log_{b}b=1; \log_{b}1=0\]
\[\log_{b}b^{x}=x\]
\[b^{\log_{b}x}=x\]
\[\log_{a}x=\frac{ \log_{b}x }{ \log_{b}a }\]
\[\log_{b}(x^{r})=rlog_{b}x\]
\[\log_{b}(xy)=\log_{b}x+\log_{b}y\]
\[\log_{b}\frac{ x }{ y }=\log_{b}x-\log_{b}y\]