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Chapter 8: Problem 58

Simplify. $$\sqrt{2 n}(\sqrt{6 n}+3)$$

Short Answer

Expert verified
The simplified form is \(2\root{3}路n + 3\root{2n}\).

Step by step solution

01

- Distribute the square root

Distribute \(\root{2n}\) across the terms inside the parenthesis. This means multiplying \(\root{2n}\) by each term inside the parenthesis: \(\root{2n}路(\root{6n} + 3) = \root{2n}路\root{6n} + \root{2n}路3\).
02

- Simplify the Product of Square Roots

Combine \(\root{2n}\) and \(\root{6n}\) using the property of square roots that \(\root a \cdot \root b = \root{ab}\). This gives us \(\root{2n}路\root{6n} = \root{(2n)(6n)} = \root{12n^2}\).
03

- Simplify the Square Root

Since \(n^2\) under the square root sign simplifies to \(n\): \(\root{12n^2} = \root{12}路n = 2\root{3}路n\).
04

- Final Distribution and Simplification

Combine this result with the remaining term: \(2\root{3}路n + 3\root{2n}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
When simplifying expressions like \(\root{2n}(\root{6n}+3)\), one key technique is the distributive property. This property allows you to distribute or multiply terms outside the parenthesis by each term inside the parenthesis.
For this problem, \(\root{2n}\) is distributed across \(\root{6n}+3\).
This results in:
\(\root{2n} \times \root{6n} + \root{2n} \times 3\).
Understanding the distributive property is crucial because it breaks down more complex expressions into manageable pieces.
Square Root Multiplication
Combining square roots involves a special rule: \(\root{a} \times \root{b} = \root{ab}\). This rule simplifies our expression.
In our problem, we multiply \(\root{2n} \times \root{6n}\).
According to the rule:
\(\root{2n} \times \root{6n} = \root{(2n) \times (6n)} = \root{12n^2}\).
Understanding this property is essential for moving forward.
Simplification of Square Roots
Simplifying square roots involves breaking down the expression under the root sign to its simplest form.
For \(\root{12n^2}\), this becomes simpler by recognizing \(\root{n^2} = n\).
Therefore:
\(\root{12n^2} = \root{12} \times \root{n^2} = \root{12} \times n = 2\root{3} \times n\).
Recognizing these simplifications helps make the expression clearer and easier to handle.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations. Simplifying these expressions, like \(\root{2n}(\root{6n}+3)\), involves applying properties and rules to combine and reduce terms.
After simplifying the square roots and distributing, our final expression is:
\(\root{2n} \times \root{6n} + \root{2n} \times 3 = 2\root{3} \times n + 3\root{2n}\).
Working through these steps ensures a clear and simplified final result, making algebraic expressions more manageable and less intimidating.

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Most popular questions from this chapter

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The completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Simplify: \(\frac{x^{\frac{9}{10}}}{x^{\frac{3}{10}}}\)
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