Question
Write the reciprocal $5+\sqrt2.$
✓
Answer
Given that $5+\sqrt2,$ it's reciprocal is given as$\frac{1}{5+\sqrt2}$
It can be simplified by rationalizing the denominator. The rationalizing factor of $5+\sqrt2$ is $5-\sqrt2,$ we will multiply numerator and denominator of the given expression $\frac{1}{5+\sqrt2}$ by $5-\sqrt2,$ to get$\frac{1}{5+\sqrt2}\times\frac{5-\sqrt2}{5-\sqrt2}=\frac{5-\sqrt2}{(5)^2\big(\sqrt2\big)^2}$
$=\frac{5-\sqrt2}{25-2}$
$=\frac{5-\sqrt2}{23}$
Hence reciprocal of the given expression is $\frac{5-\sqrt2}{23}.$
It can be simplified by rationalizing the denominator. The rationalizing factor of $5+\sqrt2$ is $5-\sqrt2,$ we will multiply numerator and denominator of the given expression $\frac{1}{5+\sqrt2}$ by $5-\sqrt2,$ to get$\frac{1}{5+\sqrt2}\times\frac{5-\sqrt2}{5-\sqrt2}=\frac{5-\sqrt2}{(5)^2\big(\sqrt2\big)^2}$
$=\frac{5-\sqrt2}{25-2}$
$=\frac{5-\sqrt2}{23}$
Hence reciprocal of the given expression is $\frac{5-\sqrt2}{23}.$
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