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Chapter 3: Problem 62

Show that \(\sqrt{2+\sqrt{3}}=\sqrt{\frac{3}{2}}+\sqrt{\frac{1}{2}}\).

Short Answer

Expert verified
After squaring both sides of the equation and simplifying, we find that both sides are equal: \((\sqrt{2+\sqrt{3}})^2 = 2+\sqrt{3}\). Therefore, the initial equation is true and we have shown that \(\sqrt{2+\sqrt{3}}=\sqrt{\frac{3}{2}}+\sqrt{\frac{1}{2}}\).

Step by step solution

01

Square both sides of the equation

We will square both sides of the equation, which will eliminate the square roots: \((\sqrt{2+\sqrt{3}})^2 = (\sqrt{\frac{3}{2}}+\sqrt{\frac{1}{2}})^2\)
02

Expand the right side of the equation

In this step, we will expand the right side of the equation using the distributive property, we get that: \((\sqrt{\frac{3}{2}}+\sqrt{\frac{1}{2}})^2 = (\sqrt{\frac{3}{2}})^2 + 2(\sqrt{\frac{3}{2}})(\sqrt{\frac{1}{2}}) + (\sqrt{\frac{1}{2}})^2\)
03

Simplify the right side of the equation

Simplify each term to get that: \((\sqrt{\frac{3}{2}})^2 = \frac{3}{2}\), \(2(\sqrt{\frac{3}{2}})(\sqrt{\frac{1}{2}}) = 2\sqrt{\frac{3}{4}} = \sqrt{3}\), and \((\sqrt{\frac{1}{2}})^2 = \frac{1}{2}\) Therefore, the right side of the equation becomes: \[\frac{3}{2}+\sqrt{3}+\frac{1}{2}\]
04

Simplify the right side further

Combine the like terms to get that: \[\frac{3}{2}+\sqrt{3}+\frac{1}{2} = 2+\sqrt{3}\]
05

Compare the left and right sides of the equation

Now we have: \((\sqrt{2+\sqrt{3}})^2 = 2+\sqrt{3}\] Since the left side and the right side of the equation are equal, it means that the initial equation is true, and we have shown that: \[\sqrt{2+\sqrt{3}}=\sqrt{\frac{3}{2}}+\sqrt{\frac{1}{2}}\]

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