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Chapter 9: Problem 29

Simplify the expression. $$\sqrt{18 u}+3 \sqrt{8 u}$$

Short Answer

Expert verified
The simplified form of the expression \(\sqrt{18 u}+3 \sqrt{8 u}\) is \(12\sqrt{2 u}\)

Step by step solution

01

Simplify each square root

The given expression is \(\sqrt{18 u}+3 \sqrt{8 u}\). Let's simplify each square root separately. We need to find the square root of 18 and 8 and simplify it. \(\sqrt{18 u}\) can be simplified as \(3\sqrt{2 u}\) and \(3\sqrt{8 u}\) can be simplified as \(9\sqrt{2 u}\).
02

Combine Like terms

Now we have two terms with the same square root, we can combine the coefficients. So \(3\sqrt{2 u} + 9\sqrt{2 u} = 12\sqrt{2 u}\)

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Key Concepts

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

Understanding Algebraic Expressions
Algebraic expressions are fundamental components in algebra that consist of constants, variables, and operations. When working with these expressions, it鈥檚 important to distinguish between each element. Here鈥檚 a quick breakdown:
  • Constants: These are numbers on their own, like 2, 3, or 5.
  • Variables: Usually represented by letters (like \( u \)), they signify unknown values.
  • Operations: Additions, subtractions, multiplications, or divisions that connect constants and variables.
In our example, \( \sqrt{18u} + 3\sqrt{8u} \), we identify that it contains variables under radicals (square roots) and numerical coefficients that will undergo operations like addition and multiplication.
Combining Like Terms
Combining like terms is a process of simplifying algebraic expressions by grouping similar components. This makes expressions easier to work with. Like terms are terms that contain the same variables raised to the same power.
In the expression \( 3\sqrt{2u} + 9\sqrt{2u} \), both terms are like because they have \( \sqrt{2u} \) with them. The rule for combining is to add their coefficients. For example:
  • \( 3\sqrt{2u} \) means there are three of these \( \sqrt{2u} \) terms.
  • \( 9\sqrt{2u} \) means there are nine more.
  • When combined, you simply add: \( 3 + 9 = 12 \).
Thus, the simplified expression becomes \( 12\sqrt{2u} \). It's like adding apples to apples in simple arithmetic.
Working with Radical Expressions
Radical expressions involve roots, most commonly square roots. These expressions need special attention when simplifying or solving equations.
Simplifying radical expressions is about breaking down the number into its prime factors. For example:
  • \( \sqrt{18} \) breaks down as \( \sqrt{9 \times 2} = 3\sqrt{2} \).
  • \( \sqrt{8} \) simplifies to \( \sqrt{4 \times 2} = 2\sqrt{2} \).
When variables are involved, you pair them with these coefficients, such as \( \sqrt{18u} = 3\sqrt{2u} \). When you encounter different radical terms, check if they can be simplified to have the same base. If they do, combine them, as shown in the previous section.

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