What is the value of $$\left(\sqrt{7+\sqrt{29}}-\sqrt{7-\sqrt{29}}\right)^2?$$ $$A.-26$$ $$B.\ 2\sqrt{29}$$ $$C.\ 14-4\sqrt{5}$$ $$D.\ 14$$ $$E.\ 14+4\sqrt{5}$$
The OA is C .
Experts, may you help me here and tell me how to find the correct answer? Please.
What is the value of
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Hi M7MBA,What is the value of $$\left(\sqrt{7+\sqrt{29}}-\sqrt{7-\sqrt{29}}\right)^2?$$ $$A.-26$$ $$B.\ 2\sqrt{29}$$ $$C.\ 14-4\sqrt{5}$$ $$D.\ 14$$ $$E.\ 14+4\sqrt{5}$$
The OA is C .
Experts, may you help me here and tell me how to find the correct answer? Please.
Let's take a look at your questions.
$$\left(\sqrt{7+\sqrt{29}}-\sqrt{7-\sqrt{29}}\right)^2 ... (i)$$
To evaluate it we will be using the formula:
$$\left(a-b\right)^2=a^2-2ab+b^2$$
Using this formula (i) can be written as:
$$=\left(\sqrt{7+\sqrt{29}}\right)^2-2\left(\sqrt{7+\sqrt{29}}\right)\left(\sqrt{7-\sqrt{29}}\right)+\left(\sqrt{7-\sqrt{29}}\right)^2$$
$$=\left(7+\sqrt{29}\right)-2\left(\sqrt{\left(7+\sqrt{29}\right)\left(7-\sqrt{29}\right)}\right)+\left(7-\sqrt{29}\right)$$ $$=\left(7+\sqrt{29}\right)-2\left(\sqrt{\left(7\right)^2-\left(\sqrt{29}\right)^2}\right)+\left(7-\sqrt{29}\right)$$
$$=\left(7+\sqrt{29}\right)-2\left(\sqrt{49-29}\right)+\left(7-\sqrt{29}\right)$$
$$=\left(7+\sqrt{29}\right)-2\left(\sqrt{20}\right)+\left(7-\sqrt{29}\right)$$
Combining like terms:
$$=7+7-\sqrt{29}+\sqrt{29}-2\left(\sqrt{20}\right)$$
$$=7+7-2\left(\sqrt{20}\right)$$
$$=14-2\left(\sqrt{20}\right)$$
$$=14-2\left(\sqrt{4\times5}\right)$$
$$=14-2\times2\sqrt{5}$$
$$=14-4\sqrt{5}$$
Therefore, Option C is correct.
Hope it helps.
I am available if you'd like any follow up.
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We know that (x - y)^2 = (x - y)(x - y) = x^2 + y^2 - 2xy. Let’s compare that to the given expression in the question.
We see that we have an expression whose expansion will be in the form of x^2 + y^2 - 2xy, so we can let x = √(7 + √29) and y = √(7 - √29). So, we have:
x^2 = 7 + √29
y^2 = 7 - √29
2xy = 2[√(7 + √29)][√(7 - √29)] = 2√[7^2 - (√29)^2] = 2√(49 - 29) = 2√20 = 2(2√5) = 4√5
Thus, the final value is:
7 + √29 + 7 - √29 - 4√5 = 14 - 4√5
Answer: C
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