Question

$\sqrt{16 + 6 \sqrt{7} }$
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Answer 1

$\huge{\underline{\underline{\mathfrak{Answer}}}}$

111−

16+6+

7

111−

16+6+2.65

111−4.96

=106.04

Please mark as brilliant answer.

$\huge{\underline{\underline{\mathfrak{Thank You}}}}$

Answer 2

Answer:

$\sqrt{16+6\sqrt{7}}=\sqrt{7}+3\quad \left(\mathrm{Decimal:\quad }\:5.64575\dots \right)$

Step-by-step explanation:

$\sqrt{16+6\sqrt{7}}$

$\black{\mathrm{Factor}\:16+6\sqrt{7}:}$

$16+6\sqrt{7}$

$\gray{\mathrm{Add/Subtract\:}\sqrt{7}^2=7}$

$=16+6\sqrt{7}+\sqrt{7}^2-7$

$\gray{\mathrm{Refine}}$

$=\sqrt{7}^2+6\sqrt{7}+9$

$\gray{\mathrm{Rewrite\:}\sqrt{7}^2+6\sqrt{7}+9\mathrm{\:as\:}\sqrt{7}^2+2\sqrt{7}\cdot \:3+3^2}$

$=\sqrt{7}^2+2\sqrt{7}\cdot \:3+3^2$

$\gray{\mathrm{Apply\:Perfect\:Square\:Formula}:\quad \left(a+b\right)^2=a^2+2ab+b^2}$

$\gray{a=\sqrt{7},\:b=3}$

$=\left(\sqrt{7}+3\right)^2$

$=\sqrt{\left(\sqrt{7}+3\right)^2}$

$\gray{\mathrm{Apply\:radical\:rule\:}\sqrt[n]{a^n}=a,\:\quad \mathrm{\:assuming\:}a\ge 0}$

$=\sqrt{7}+3$