Problem Solving (PS) | Re: What is the value of (\sqrt{4-\sqrt{15}})(4 + \sqrt{15})

OfficialSolution:

What is the value of\( (\sqrt{4-\sqrt{15}})(4 + \sqrt{15})(\sqrt{6} -\sqrt{10})\) ?

A.\(-4\)
B.\(-2\)
C.\(-1\)
D.\(1\)
E.\(2\)


We want to find the value of the expression\( (\sqrt{4-\sqrt{15}})(4 + \sqrt{15})(\sqrt{6} -\sqrt{10})\) . To simplify this expression, we can start by squaringit.

\( =((\sqrt{4-\sqrt{15}})(4 + \sqrt{15})(\sqrt{6} -\sqrt{10}))^2=\)

\( =(4-\sqrt{15})(4 + \sqrt{15})^2(\sqrt{6} -\sqrt{10})^2=\)

\( =(4-\sqrt{15})(4 + \sqrt{15})(4 + \sqrt{15})(\sqrt{6} -\sqrt{10})^2=\)

Applying the difference of squares identity\( (a -b)(a+b)=a^2-b^2\) to\( (4-\sqrt{15})(4 +\sqrt{15})\) we can simplifyto:

\( =(16 -15)(4 + \sqrt{15})(\sqrt{6} -\sqrt{10})^2=\)

Now using the identity\( (a -b)^2=a^2-2ab+b^2\) to simplify the squaredterm::

\( =(4 + \sqrt{15})(6 -2\sqrt{60}+10)=\)

\( =(4 + \sqrt{15})(16 -4\sqrt{15})=\)

\( =(4 + \sqrt{15})4(4 -\sqrt{15})=\)
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Statistics : Posted by Bunuel • on 04 Sep 2022, 09:35 • Replies 2 • Views 1032

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