Question

Please simplify the radical

Answer 1

Check the picture below.

`\begin{array}{llll} \textit{using the pythagorean theorem} \\\\ c^2=a^2+o^2\implies c=√(a^2 + o^2) \end{array} \qquad \begin{cases} c=\stackrel{hypotenuse}{y}\\ a=\stackrel{adjacent}{2}\\ o=\stackrel{opposite}{4} \end{cases} \\\\\\ y=√( 2^2 + 4^2)\implies y=√( 4 + 16 ) \implies y=√( 20 ) \\\\[-0.35em] ~\dotfill`

`\begin{array}{llll} \textit{using the pythagorean theorem} \\\\ a^2+y^2=c^2\implies a=√(c^2 - y^2) \end{array} \qquad \begin{cases} c=\stackrel{hypotenuse}{10}\\ a=\stackrel{adjacent}{x}\\ y=\stackrel{opposite}{√(20)} \end{cases} \\\\\\ x=\sqrt{ 10^2 - (√(20))^2}\implies x=√( 100 - 20 ) \implies x=√( 80 )\implies x=4√(5)`