Question

\(\sqrt{3+\sqrt{5}}=\) (1) \(\sqrt{2}+1\) (2) \(\sqrt{\frac{5}{2}}+\sqrt{\frac{1}{2}}\) (3) \(\sqrt{\frac{7}{2}}-\sqrt{\frac{1}{2}}\) (4) \(\sqrt{\frac{9}{2}}-\sqrt{\frac{3}{2}}\)

Short answer

A) \(\sqrt{2}+1\) B) \(\sqrt{\frac{5}{2}}+\sqrt{\frac{1}{2}}\) C) \(\sqrt{\frac{7}{2}}-\sqrt{\frac{1}{2}}\) D) \(\sqrt{\frac{9}{2}}-\sqrt{\frac{3}{2}}\) Answer: None of the provided options are equal to the given expression \(\sqrt{3+\sqrt{5}}\).

Step-by-step solution

Simplify the given expression

The given expression is \(\sqrt{3+\sqrt{5}}\). Since there are no common terms to combine or simplify, we will leave this expression as is.

Simplify Option 1

Option 1 is \(\sqrt{2}+1\). Since there are no nested square roots, we can compare this option to the given expression. The given expression has a nested square root, but this option does not, so they are not equal.

Simplify Option 2

Option 2 is \(\sqrt{\frac{5}{2}}+\sqrt{\frac{1}{2}}\). To compare this expression to the given expression, we can first try to combine the two square roots: \(\sqrt{\frac{5}{2}}+\sqrt{\frac{1}{2}} = \sqrt{(\frac{5+1}{2})} = \sqrt{\frac{6}{2}} = \sqrt{3}\) This expression is not equal to the given expression, so Option 2 does not match the given expression.

Simplify Option 3

Option 3 is \(\sqrt{\frac{7}{2}}-\sqrt{\frac{1}{2}}\). Similar to Option 2, we will try to combine the two square roots: \(\sqrt{\frac{7}{2}}-\sqrt{\frac{1}{2}} = \sqrt{(\frac{7-1}{2})} = \sqrt{\frac{6}{2}} = \sqrt{3}\) This expression is not equal to the given expression, so Option 3 does not match the given expression.

Simplify Option 4

Option 4 is \(\sqrt{\frac{9}{2}}-\sqrt{\frac{3}{2}}\). Again, we will try to combine the two square roots: \(\sqrt{\frac{9}{2}}-\sqrt{\frac{3}{2}} = \sqrt{(\frac{9-3}{2})} = \sqrt{\frac{6}{2}} = \sqrt{3}\) This expression is not equal to the given expression, so Option 4 does not match the given expression. None of the provided options are equal to the given expression \(\sqrt{3+\sqrt{5}}\).

Key concepts

Radicals Simplification

Simplifying radical expressions, commonly known as square roots, involves several steps to reduce the expression to its simplest form. In our example, we're given a nested square root, \(\sqrt{3+\sqrt{5}}\), which might seem daunting at first glance. However, to simplify radicals, we focus on identifying perfect squares within the radical, which can be taken out of the square root as their square root values.

Consider the expression \(\sqrt{n + \sqrt{m}}\). To simplify, you would look for square numbers that closely relate to 'n' and 'm' to see if they can be expressed as such. If 'n' and 'm' are not perfect squares, however, and they don't contain any perfect square factors, further simplification without approximation isn't possible.

In cases like our exercise, because there are no perfect squares within the nested square root that we can simply take out, we cannot simplify the expression any further.

Comparing Square Roots

When comparing square roots, it's important to ascertain whether the expressions represent the same quantity. This can sometimes involve simplifying the square roots or trying to determine if one square root is the sum or difference of others. In the textbook problem we are addressing, we compare the given nested square root with several options, looking for an equivalent expression.

Common Misconceptions

Many students try to apply operations such as addition or subtraction within different levels of nested square roots, expecting to simplify them directly. However, unless the nested square roots can be combined or simplified into perfect squares, these operations will not yield correct comparisons or simplifications. In our example, the given expression remains distinct because none of the options reflects the nested square root structure it contains.

Square Root Operations

Performing operations with square roots involves rules that are slightly different from those for regular numbers. Adding or subtracting square roots is only possible when the radicands (the numbers under the square root sign) are identical. Otherwise, the square roots are considered 'like terms.' Similarly, multiplying and dividing square roots follow certain rules where you can multiply or divide the radicands and then take the square root of the result.

In our textbook problem, the solutions attempted to add or subtract square roots with different radicands. This is an incorrect approach unless a common radicand can be found or created. Since this was not achieved in our options and square root expressions were incorrectly combined, none of the operations provided a correct match to the original expression \(\sqrt{3+\sqrt{5}}\).

Exercise Improvement Advice

To improve understanding, always remember that operations on square roots must respect the unique properties of radicals. When dealing with nested square roots, it's imperative to explore whether it's possible to simplify them separately before attempting any direct operations.