Answer
$26-2\sqrt{105}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given radical expression, $
(\sqrt{21}-\sqrt{5})^2
,$ use the special product on squaring binomials and the properties of radicals. Then combine like terms.
$\bf{\text{Solution Details:}}$
Using the square of a binomial which is given by $(a+b)^2=a^2+2ab+b^2$ or by $(a-b)^2=a^2-2ab+b^2,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
(\sqrt{21})^2-2(\sqrt{21})(\sqrt{5})+(\sqrt{5})^2
\\\\=
21-2(\sqrt{21})(\sqrt{5})+5
.\end{array}
Using the Product Rule of radicals which is given by $\sqrt[m]{x}\cdot\sqrt[m]{y}=\sqrt[m]{xy},$ the expression above is equivalent to\begin{array}{l}\require{cancel}
21-2\sqrt{21(5)}+5
\\\\=
21-2\sqrt{105}+5
.\end{array}
By combining like terms, the expression above is equivalent to
\begin{array}{l}\require{cancel}
(21+5)-2\sqrt{105}
\\\\=
26-2\sqrt{105}
.\end{array}
