Question

What is the exact value of the expression the square root of 20. + the square root of 24. - the square root 54.? simplify if possible?

Answer 1

.  $\sqrt{20}+ \sqrt{24} - \sqrt{54}$.  The idea is to simplify the radicand into a perfect square or a multiplication of 2 numbers and 1 of those is a perfect square.  Let's rewrite our simplified radicands: $\sqrt{4*5} + \sqrt{4*6}- \sqrt{9*6}$.  In the first two, 4 is the perfect square and will be pulled out as a 2, leaving whatever is left under the radical sign, and in the third one, 9 is the perfect square that will be pulled out as a 3.  $2 \sqrt{5} +2 \sqrt{6} -3 \sqrt{6}$.  The rule for adding and subtracting radicands is that a. the index has to be the same (ours are all square roots, so the indexes are all 2), and b. the radicands have to be like in order to combine them.  Our square root of 6 is in 2 of those, so those can be combined, but not the square root of 5.  Combining like terms now we have $2 \sqrt{5} -1 \sqrt{6}$

Answer 2

$\sqrt{20} = \sqrt{4} \sqrt{5} = 2 \sqrt{5}$
$\sqrt{24} = \sqrt{4} \sqrt{6} = 2 \sqrt{6}$
$\sqrt{54} = \sqrt{9} \sqrt{6} = 3 \sqrt{6}$

Because you can only add and subtract radicals with the same radicand,
you get 2$\sqrt{5} - \sqrt{6}$
that is the simplified answer but if you want it in decimals, 2.022 is what you get.