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Chapter 15: Problem 43

Simplify. $$4 \sqrt{2}-5 \sqrt{2}+8 \sqrt{2}$$

Short Answer

Expert verified
The simplified form of the expression is \(7\sqrt{2}\).

Step by step solution

01

Identify like terms

The terms \(4 \sqrt{2}\), \(-5 \sqrt{2}\) and \(8 \sqrt{2}\) are seen as like terms. This is because their radical parts, or the parts involving square roots, are the same (\(\sqrt{2}\)) for all of them.
02

Combine like terms

Like terms can be combined by adding or subtracting the coefficients (the numbers in front of the square roots). The calculation that results from this is \(4\sqrt{2} - 5\sqrt{2} + 8\sqrt{2}\).
03

Perform the Addition/Subtraction

When you perform the addition/subtraction, you subtract 5 from 4 and then add 8: \(4 - 5 + 8 = 7\). You keep the radical part of the term as is. As a result, you get \(7\sqrt{2}\).

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Key Concepts

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

Like Terms
When simplifying mathematical expressions, identifying like terms is essential. Like terms are terms that contain the same variables raised to the same power. In the context of radicals, like terms must have the same radical part. For example, in the expression given, \(4 \sqrt{2}\), \(-5 \sqrt{2}\), and \(8 \sqrt{2}\) are like terms because they all have the same radical part, \(\sqrt{2}\).

Recognizing like terms allows you to combine them through addition or subtraction. The idea here is that you can treat the coefficient of each term as if you were dealing with simple numbers. This is because the radical part is the same, so it acts as a common factor across the terms.
  • It simplifies expressions and makes them easier to work with.
  • Helps in solving algebraic equations.
Radicals
Radicals are expressions that include a root symbol, typically a square root. They can sometimes appear complex, but they operate under specific rules that simplify calculations. When we refer to radicals, we're discussing the portion of an expression that is under the radical sign, in this case, \(\sqrt{2}\).

It's important to understand that while you can't directly add or subtract radicals with different radicands (the number or expression inside the root), radicals with the same radicand can be combined by focusing on their coefficients. Always remember that the process involves manipulating the coefficients while keeping the radical part unchanged.
  • Allows transformations from an expression under a radical to find equivalent forms.
  • Requires practice to become efficient in identifying and simplifying them.
Coefficients
Coefficients are the numerical factors in front of terms. In the example \(4 \sqrt{2} - 5 \sqrt{2} + 8 \sqrt{2}\), \(4\), \(-5\), and \(8\) are coefficients. They determine how many times the radical part is taken in the expression.

When combining like terms, you only combine the coefficients. It's akin to adding or subtracting numbers, while the radical portion remains constant between the terms. This means in our case, you will do: \(4 - 5 + 8\), which results in \(7\). The result is then expressed with the unchanged radical as \(7 \sqrt{2}\).
  • Vital in maintaining the structure of expressions when simplifying radicals.
  • Understanding them is key to performing algebraic manipulations accurately.

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