# How to simplify a diabolical expression involving radicals

• MHB
• kalish1
In summary, the conversation discusses simplifying a complex expression involving radicals. The expression equals 1/2 and several substitutions have been attempted, but none have been successful. A potential solution is suggested involving the substitution of $u = \sqrt{5 + 2 \sqrt{5}}$ to simplify the fraction.
kalish1
A friend and I have been working on this problem for hours - how can the following expression be simplified analytically?

It equals $\frac{1}{2},$ and we have tried the following to no avail:

1. Substitution of $x = \sqrt{5}$
2. Substitution of $x = 2\sqrt{5}$
3. Substitution of $x = 5+\sqrt{5}$
4. Substitution of $x = \sqrt{5 + \sqrt{5}}$

Here goes:
$$\dfrac{\dfrac{\sqrt{5 + 2\sqrt{5}}}{2} + \dfrac{\sqrt{5(5 + 2\sqrt{5})}}{4} - \dfrac{\sqrt{10 + 2\sqrt{5}}}{8}}{\dfrac{\sqrt{5(5 + 2\sqrt{5})}}{4} + 5 \cdot \dfrac{\sqrt{5 + 2\sqrt{5}}}{4}}$$

Thanks in advance for any help.

This question has been crossposted here - fractions - How to simplify a diabolical expression involving radicals - Mathematics Stack Exchange

Try the substitution $u = \sqrt{5 + 2 \sqrt{5}}$. Then you have:
$$\sqrt{5(5 + 2 \sqrt{5})} = \sqrt{5} u$$
$$\sqrt{10 + 2 \sqrt{5}} = \sqrt{u^2 + 5}$$
You'll still have a root but you'll be able to simplify the fraction I think.

## What is a diabolical expression involving radicals?

A diabolical expression involving radicals is an expression that contains square roots, cube roots, or higher-order roots, as well as variables and numerical constants. These expressions can be complex and challenging to simplify.

## What are some strategies for simplifying diabolical expressions involving radicals?

One strategy is to identify perfect square factors and simplify them. Another strategy is to rationalize the denominator by multiplying by the conjugate. Additionally, you can use the properties of exponents to simplify expressions with radicals.

## How do I simplify a diabolical expression with multiple radicals?

When dealing with multiple radicals, you can simplify each radical separately and then combine the simplified terms. You can also use the distributive property to simplify expressions with multiple radicals.

## What are some common mistakes to avoid when simplifying diabolical expressions involving radicals?

A common mistake is to forget to simplify perfect square factors. Another mistake is to confuse the order of operations and incorrectly simplify the expression. It is also important to be careful when combining like terms with different radicals.

## Can diabolical expressions involving radicals be simplified to a single term?

In some cases, diabolical expressions can be simplified to a single term, but this is not always possible. It depends on the specific expression and the strategies used to simplify it. It is important to check your work and make sure the simplified expression is equivalent to the original one.

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