If \sqrt{28 - 10\sqrt{3}} + \sqrt{7 + 4\sqrt{3}} = k, where k is a rational number, find t… — JAMB UTME

If $\sqrt{28 - 10\sqrt{3}} + \sqrt{7 + 4\sqrt{3}} = k$ , where $k$ is a rational number, find the value of $k$ .

Answer: 7

The value of

k

7

Simplifying each nested surd into standard binomial surd form gives

\sqrt{28 - 10\sqrt{3}} = 5 - \sqrt{3}

and

\sqrt{7 + 4\sqrt{3}} = 2 + \sqrt{3}

. Combining these terms cancels out the irrational component

\sqrt{3}

, leaving the integer result

5 + 2 = 7

Simplify the nested surd

\sqrt{28 - 10\sqrt{3}}

into binomial surd form

\sqrt{28 - 10\sqrt{3}} = 5 - \sqrt{3}

Assuming a form

x - y\sqrt{3}

and squaring both sides gives

x^2 + 3y^2 - 2xy\sqrt{3} = 28 - 10\sqrt{3}

. Matching components leads to

xy = 5

and

x^2 + 3y^2 = 28

, which yields integer values

x = 5

and

y = 1

Simplify the nested surd

\sqrt{7 + 4\sqrt{3}}

into binomial surd form

\sqrt{7 + 4\sqrt{3}} = 2 + \sqrt{3}

Assuming a form

u + v\sqrt{3}

and squaring both sides gives

u^2 + 3v^2 + 2uv\sqrt{3} = 7 + 4\sqrt{3}

. Matching components leads to

uv = 2

and

u^2 + 3v^2 = 7

, which yields integer values

u = 2

and

v = 1

Sum the simplified expressions to calculate

k

k = 7

Summing

(5 - \sqrt{3}) + (2 + \sqrt{3})

results in the irrational parts

-\sqrt{3}

and

\sqrt{3}

cancelling out, leaving

5 + 2 = 7

Square Root of a Binomial Surd Expression

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