Find the square root of 30-2 sqrt{56}. | vedclass

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Question

(C) To find the square root of $30-2 \sqrt{56}$,we express it in the form $(a-b)^2 = a^2+b^2-2ab$.We need to find two numbers such that their sum of s

Vedclass · Accepted Answer

$2 \sqrt{7}-\sqrt{2}$

Vedclass · Answer

(C) To find the square root of $30-2 \sqrt{56}$,we express it in the form $(a-b)^2 = a^2+b^2-2ab$.We need to find two numbers such that their sum of squares is $30$ and their product is $\sqrt{56}$.$30-2 \sqrt{56} = 30-2 \sqrt{28 	imes 2} = 30-2 \sqrt{7 	imes 8}$.We can write $30$ as $(28+2)$ or $(7+23)$,but looking at the factors of $56$,we have $56 = 28 	imes 2$ or $7 	imes 8$.Let the expression be $(\sqrt{x}-\sqrt{y})^2 = x+y-2 \sqrt{xy}$.Here,$x+y = 30$ and $xy = 56$.Solving for $x$ and $y$,we find $x=28$ and $y=2$.Thus,$30-2 \sqrt{56} = 28+2-2 \sqrt{28 	imes 2} = (\sqrt{28}-\sqrt{2})^2$.Since $\sqrt{28} = \sqrt{4 	imes 7} = 2 \sqrt{7}$,the expression becomes $(2 \sqrt{7}-\sqrt{2})^2$.Therefore,the square root is $2 \sqrt{7}-\sqrt{2}$.

Find the square root of $30-2 \sqrt{56}$.

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