A problem combining a special product formula, a fractional exponent and the cosine of an obtuse angle. Solved step by step with the supporting formulas noted in the margin.

The expression looks intimidating because the roots are nested, but it breaks into three familiar steps: the square of a difference, the product of a sum and a difference under a common root, and the value of the cosine of 120°. After each one, less is left.

Calculate

\left(\left(\sqrt{13} + 2\right)^{\tfrac{1}{2}} - \left(\sqrt{13} - 2\right)^{\tfrac{1}{2}}\right)^{2} \cdot \cos \tfrac{2}{3}\pi

Solution

\left[\left(\left(\sqrt{13} + 2\right)^{\tfrac{1}{2}}\right)^{2} - 2\left(\left(\sqrt{13} + 2\right)\left(\sqrt{13} - 2\right)\right)^{\tfrac{1}{2}} + \left(\left(\sqrt{13} - 2\right)^{\tfrac{1}{2}}\right)^{2}\right] \cdot \cos \tfrac{2}{3}\pi

(a - b)^{2} = a^{2} - 2ab + b^{2}

a^{n} \cdot b^{n} = (a \cdot b)^{n}

\left(\sqrt{13} + 2 - 2(13 - 4)^{\tfrac{1}{2}} + \sqrt{13} - 2\right) \cdot \cos \tfrac{2}{3}\pi

\left(a^{\tfrac{1}{2}}\right)^{2} = a^{\tfrac{1}{2} \cdot 2} = a^{1} = a

(a + b)(a - b) = a^{2} - b^{2}

\left(2\sqrt{13} - 2 \cdot 9^{\tfrac{1}{2}}\right) \cdot \cos 120^{\circ}

\left(2\sqrt{13} - 6\right) \cdot \left(-\tfrac{1}{2}\right)

-\sqrt{13} + 3 = 3 - \sqrt{13}

Notes in the margin

\pi = 180^{\circ}

radians to degrees

\tfrac{2}{3} \cdot 180^{\circ} = 120^{\circ}

\cos 120^{\circ} = \cos (90^{\circ} + 30^{\circ}) = -\sin 30^{\circ} = -\tfrac{1}{2}

the cosine of an obtuse angle

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Frequently asked questions

Why can a product of roots be written under a single root?

Because powers with the same exponent multiply by multiplying the bases: a to the n times b to the n equals a times b, all to the n. A root is a power with a fractional exponent, so the same rule applies.

What is the cosine of 120 degrees?

Minus one half. Write 120° as 90° plus 30°; the cosine of that sum is minus the sine of 30 degrees, which is minus one half.

How do you convert radians to degrees?

Just substitute 180 degrees for π. Two thirds of π is two thirds times 180 degrees, which is 120 degrees.

A lesson from the PhiBoard knowledge base: phiboard.com/en/knowledge-base/interesting-math-problems/nested-roots-and-cosine-of-120-degrees. On the lesson page you will also find games and practice exercises - free and without an account.

A fraction with powers, a squared logarithm and the sine of 210°