What is the simplified value of \sqrt{7 + 4\sqrt{3}} - \sqrt{3}? — State PSC Exam

What is the simplified value of $\sqrt{7 + 4\sqrt{3}} - \sqrt{3}$ ?

Answer: 2

The simplified value is 2.

Expressing

7 + 4\sqrt{3}

(2 + \sqrt{3})^2

allows the square root to simplify directly to

2 + \sqrt{3}

. Subtracting

\sqrt{3}

leaves the exact numerical answer 2.

Rewrite the expression under the square root as a perfect square of a binomial.

7 + 4\sqrt{3} = 2^2 + (\sqrt{3})^2 + 2(2)(\sqrt{3}) = (2 + \sqrt{3})^2

Using the identity

(a+b)^2 = a^2 + b^2 + 2ab

, setting

a = 2

and

b = \sqrt{3}

yields

a^2 + b^2 = 4 + 3 = 7

and

2ab = 4\sqrt{3}

Evaluate the square root of the perfect square.

\sqrt{(2 + \sqrt{3})^2} = 2 + \sqrt{3}

The principal square root of a positive squared expression

\sqrt{x^2}

x

Perform the subtraction indicated in the stem.

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

The radical terms

\sqrt{3}

and

-\sqrt{3}

cancel out, leaving the integer 2.

Simplification of Nested Surds

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