Alg2.1.12

Find the value of x if $\sqrt{23+x\sqrt{10}}=\sqrt{18}+\sqrt{5}\,$

Squaring on both sides

$(\sqrt{23+x\sqrt{10}})^2=(\sqrt{18}+\sqrt{5})^2\,$

$23+x\sqrt{10}=18+5+2(\sqrt{18})(\sqrt{5})\,$

$23+x\sqrt{10}=23+2\sqrt{90}\,$

Simplifying

$x\sqrt{10}=2\sqrt{90}\,$

Squaring on bothsides we get

$(x\sqrt{10})^2=(2\sqrt{90})^2\,$

$10x^2=4(90)\,$

Hence $x^2=36\,$

$x=6\,$

The value of x is 6