Question

If $x=\sqrt{\frac{5+2\sqrt{6}}{5-2\sqrt{6}}},\text{then}{x}^2(x-10{)}^2=$

• A. $5$
• B. $5^2$
• C. $1$
• D. $2^2-1$

Answer 1

The correct option is C

$1$

$x=\sqrt{\frac{5+2\sqrt{6}}{5-2\sqrt{6}}\times \frac{5+2\sqrt{6}}{5+2\sqrt{6}}}=\sqrt{\frac{(5+2\sqrt{6}{)}^2}{(5{)}^2-(2\sqrt{6}{)}^2}}=\frac{5+2\sqrt{6}}{\sqrt{25-24}}=\frac{5+2\sqrt{6}}{1}=5+2\sqrt{6}\therefore x^2(x-10{)}^2=(5+2\sqrt{6}{)}^2(5+2\sqrt{6}-10{)}^2=(5+2\sqrt{6}{)}^2(2\sqrt{6}-5{)}^2=(25+24+20\sqrt{6}\left)\right(24+25-20\sqrt{6})=(49+20\sqrt{6}\left)\right(49-20\sqrt{6})=(49{)}^2-(20\sqrt{6}{)}^2=2401-2400=1$