What is the square root of 15-4{14}?

Correct Answer is : 22−72{2}-{7}22−7

Correct Answer is : 22−72{2}-{7}22−7

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UPSC CDS 1 2021 Math Paper

Question : 52 of 100

Marks: +1, -0

15-4\sqrt{14}

Solution:

Concept:

Express the given expression as a perfect square of a binomial difference.

Explanation:

Let

15 - 4\sqrt{14} = a

We rewrite

15

8 + 7

and

4\sqrt{14}

2 \times 2\sqrt{14} = 2 \times \sqrt{56}

Observe that

8 = (\sqrt{8})^2

7 = (\sqrt{7})^2

, and

\sqrt{56} = \sqrt{8} \times \sqrt{7}

a = (\sqrt{8})^2 + (\sqrt{7})^2 - 2(\sqrt{8})(\sqrt{7})

This matches the identity

x^2 + y^2 - 2xy = (x - y)^2

with

x = \sqrt{8}

and

y = \sqrt{7}

Hence

a = (\sqrt{8} - \sqrt{7})^2

Taking the positive square root,

\sqrt{a} = \sqrt{8} - \sqrt{7}

Since

\sqrt{8} = 2\sqrt{2}

, we get

\sqrt{15 - 4\sqrt{14}} = 2\sqrt{2} - \sqrt{7}

Answer:

Option A:

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

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