Question

Express \(\sqrt{27+\sqrt{200}}\) as \(a+\sqrt{b}\). (This problem and the next two are from Rudolf's Coss.)

Short answer

Question: Express the given expression \(\sqrt{27+\sqrt{200}}\) as \(a+\sqrt{b}\). Answer: \(\sqrt{27+\sqrt{200}}\) can be expressed as \(3+\sqrt{2}\), where \(a = 3\) and \(b = 2\).

Step-by-step solution

Write down the given expression

Given expression: \(\sqrt{27+\sqrt{200}}\).

Rewrite the expression inside the square root as a sum of two squares

We can rewrite the expression inside the square root as: \(27+\sqrt{200} = 27 + 10\sqrt{2} = 9\cdot3 + 10\sqrt{2}\).

Recognize a pattern that allows further simplification

Notice that the inner square root expression (10√2) can be written as a sum of two squares: \(\sqrt{2} = \sqrt{1+1}\). So, we can rewrite the expression inside the square root as follows: \(9\cdot3 + 10\sqrt{1^2+1^2}\). The point of writing the expression in this way is that it has a pattern that can be simplified further.

Simplify the expression

The expression inside the square root can be rewritten as: \((3+\sqrt{2})^2\). Therefore, our given expression can be written as: \(\sqrt{27+\sqrt{200}}=\sqrt{(3+\sqrt{2})^2}\).

Write the simplified expression in the desired form

Since we have a square inside the square root, we can simplify it as follows: \(\sqrt{27+\sqrt{200}}=\sqrt{(3+\sqrt{2})^2}=3+\sqrt{2}\). So, in the given form \(a+\sqrt{b}\), we get \(a=3\) and \(b=2\).

Key concepts

Simplification of expressions

Simplification of expressions is a fundamental skill in mathematics that involves reducing complex expressions into simpler forms. This makes it easier to understand and work with the expressions.
In the given exercise, we begin with the complex expression \(\sqrt{27 + \sqrt{200}}\). The goal is to represent it in a simpler form: \(a + \sqrt{b}\). This means transforming the expression so that it becomes more straightforward to interpret.
To simplify an expression:

• Look for opportunities to factorize or rearrange terms. This reduces the complexity.
• Use mathematical identities or patterns, such as squares or other notable forms, to reorganize the terms.

For example, in the solution, recognizing that \(10\sqrt{2}\) could be broken down as \(\sqrt{1^2+1^2}\) was key. This introduces a recognizable pattern, allowing further simplification. The initial complex expression \(27 + \sqrt{200}\) is thus broken down, helping us arrive at a simpler result. Mastering simplification allows for quicker problem-solving and deeper insight into mathematical expressions.

Algebraic manipulation

Algebraic manipulation is crucial for simplifying expressions and solving equations. It involves using algebra rules to reshape expressions, making them easier to work with. The objective is to transform given expressions into more usable or interpretable forms.
In the provided exercise, algebraic manipulation begins by reshaping the expression \(27 + \sqrt{200}\). Here, recognizing and rewriting \(\sqrt{200}\) as \(10\sqrt{2}\) immediately makes the expression more manageable.
Follow these steps for effective algebraic manipulation:

• Identify patterns or identities that can be utilized, such as \( (x^2 + y^2) \), which can be rewritten or factored.
• Apply algebraic identities like \((a+b)^2 = a^2 + 2ab + b^2\) where applicable.
• Perform operations consistently on both sides of an equation to maintain equality and derive at the required form.

By writing the expression \((3 + \sqrt{2})^2\), we've successfully manipulated it to align with known algebraic structures, facilitating the next steps towards the solution. This illustrates how algebraic manipulation not only simplifies but also makes expression forms recognizable.

Square roots

Square roots are a common mathematical tool used to simplify expressions and are essential for breaking down complex problems into simpler parts. Understanding how to properly handle square roots can aid in efficiently solving mathematical challenges.
In the context of the exercise, we are tasked with expressing \(\sqrt{27 + \sqrt{200}}\) in the form of \(a + \sqrt{b}\). Here, the challenge is to deconstruct the complex nested square root.
Steps for handling expressions involving square roots:

• Identify components under the square root that might allow simplification, such as perfect squares.
• Break down the inner square root to form recognizable patterns that can be expanded or reduced.
• Use algebraic manipulation in tandem with square root properties to transform expressions into simpler forms.

Recognizing \( (3 + \sqrt{2})^2 \) helped simplify \(\sqrt{27+\sqrt{200}}\) into two separate terms \(3+\sqrt{2}\), which are much easier to handle and understand. Mastery of square roots involves knowing when and how to deconstruct and manipulate them to clarify complex problems efficiently.