What is the square root of the surd expression 14 + 6\sqrt{5}? — JAMB UTME

What is the square root of the surd expression $14 + 6\sqrt{5}$ ?

3 + \sqrt{5}

Expanding the square of

3 + \sqrt{5}

yields

(3)^2 + 2(3)(\sqrt{5}) + (\sqrt{5})^2 = 9 + 6\sqrt{5} + 5 = 14 + 6\sqrt{5}

, which accurately equals the original expression under the radical.

Set up the general form for the square root of a binomial surd

Let

\sqrt{14 + 6\sqrt{5}} = \sqrt{a} + \sqrt{b}

The square root of a compound surd expression takes the form of a sum of radical terms

Square both sides of the equation

14 + 6\sqrt{5} = a + b + 2\sqrt{ab}

Eliminate the outer radical to equate real and surd parts

Equate the rational parts and the surd parts

a + b = 14

and

2\sqrt{ab} = 6\sqrt{5} \implies \sqrt{ab} = 3\sqrt{5} = \sqrt{45} \implies ab = 45

Match integer terms together and radical terms together

Solve for values of

a

and

b

Two positive numbers with sum

14

and product

45

are

9

and

5

, so

a = 9

and

b = 5

Determine the factors satisfying both equations

Substitute

a

and

b

into the radical expression

\sqrt{14 + 6\sqrt{5}} = \sqrt{9} + \sqrt{5} = 3 + \sqrt{5}

Simplify

\sqrt{9}

3

to obtain the final simplified expression

Finding the square root of a surd expression by equating rational and radical parts

Question