Square roots on square roots

Algebra Level 2

Find the value of

${\left( \sqrt { \sqrt { 81 } +\sqrt { 56 } } +\sqrt { \sqrt { 81 } -\sqrt { 56 } } \right) }^{ 2 }$

The answer is 28.

2 solutions

Romain Bouchard
Apr 18, 2018

$(\sqrt{\sqrt{81}+\sqrt{56}}+\sqrt{\sqrt{81}-\sqrt{56}})^2 = (\sqrt{\sqrt{81}+\sqrt{56}})^2 + 2\sqrt{\sqrt{81}+\sqrt{56}}\sqrt{\sqrt{81}-\sqrt{56}}+(\sqrt{\sqrt{81}+\sqrt{56}})^2$

$=(\sqrt{81}+\sqrt{56}) + 2\sqrt{(\sqrt{81}+\sqrt{56})(\sqrt{81}-\sqrt{56})}+(\sqrt{81}-\sqrt{56})$

$=2\sqrt{81} + 2\sqrt{(\sqrt{81})^2-(\sqrt{56})^2}$

$=2 \times 9 + 2\sqrt{81-56}$

$=2 \times 9 + 2\sqrt{25}$

$=2 \times 9 + 2 \times 5$

$=\boxed{28}$

I like this solution, because it's just the straightforward distribution with no tricks (which is how I calculated it). I also like XX's solution though, since it noticed something clever that I did not.

Nathan Oshlag - 3 years, 1 month ago

X X
Apr 20, 2018

$\sqrt { \sqrt { 81 } +\sqrt { 56 } } =\sqrt {(7+2)+2\sqrt{7\times2}}=\sqrt{7}+\sqrt{2}$
$\sqrt { \sqrt { 81 } -\sqrt { 56 } } =\sqrt {(7+2)-2\sqrt{7\times2}}=\sqrt{7}-\sqrt{2}$
${\left( \sqrt { \sqrt { 81 } +\sqrt { 56 } } +\sqrt { \sqrt { 81 } -\sqrt { 56 } } \right) }^{ 2 }=(\sqrt{7}+\sqrt{2}+\sqrt{7}-\sqrt{2})^{2}=(2\sqrt{7})^{2}=28$

It might be worth adding a step or two on each of the first two lines showing the following steps:

$\sqrt{7+2+2\sqrt{7x2}}$

$\sqrt{7+2+2\sqrt{7}\sqrt{2}}$

$\sqrt{\sqrt{7}^2+\sqrt{2}^2+2\sqrt{7}\sqrt{2}}$

$\sqrt{(\sqrt{7}+\sqrt{2})^2}$

Nathan Oshlag - 3 years, 1 month ago

But I really like your solution, since it noticed the trick that I missed in the problem.

Nathan Oshlag - 3 years, 1 month ago

Square roots on square roots

The answer is 28.

2 solutions

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