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Chapter 0: Problem 37

In Exercises \(33-44,\) add or subtract terms whenever possible. $$\sqrt{8}+3 \sqrt{2}$$

Short Answer

Expert verified
The simplified result of the expression \(\sqrt{8} + 3\sqrt{2}\) is \(5\sqrt{2}\).

Step by step solution

01

Simplify Square Roots

The square root of 8 can be simplified further because 8 is not a prime number. It can be written as \(2^3\), which means we can simplifiy \(\sqrt{8}\) into \(2\sqrt{2}\)
02

Addition of Like Terms

Now, we have simplified terms: \(2\sqrt{2} + 3\sqrt{2}\). As they are like terms (both are multiples of \(\sqrt{2}\)), they can be added together to give \(5\sqrt{2}\).

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Key Concepts

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

Addition of Like Terms
In algebra, simplifying expressions often involves adding terms that are alike. Like terms are terms that have the same variable raised to the same power. In the context of radicals, like terms have the same radical part.
For example, in the expression \(2\sqrt{2} + 3\sqrt{2}\), both terms share the same radical part, \(\sqrt{2}\). Therefore, these are like terms and can be combined.
  • First, rewrite the expression showing the common radical part. Here it will be written as \((2 + 3)\sqrt{2}\).
  • Next, perform the addition of coefficients. Combine them: \(2 + 3 = 5\).

So, \(2\sqrt{2} + 3\sqrt{2} = 5\sqrt{2}\). This simplification helps in reducing the expression to its simplest form. Understanding how to recognize like terms is key to mastering algebraic simplifications.
Algebraic Expressions
Algebraic expressions are made up of variables, constants, and mathematical operators such as plus, minus, multiplication, and division. Each component in an expression can participate in operations such as addition or subtraction.
These expressions often include terms, which are the building blocks connected by addition or subtraction. For instance, in an algebraic expression like \(2x + 3y - 4z\), each component separated by an operator is called a term.
  • Variables in terms (like \(x\), \(y\), and \(z\)) represent unknown values that can vary.
  • Coefficients (like 2, 3, and -4) describe how much of the variable there is.
  • Constants are standalone numbers without any variables attached, though this particular example does not have one.

Simplifying algebraic expressions often involves combining like terms and applying basic arithmetic operations to create a simpler, more compact form.
Radicals
Radicals are expressions that involve roots, most commonly square roots. The term under the root symbol is called the radicand. Understanding radicals involves simplifying them to their most manageable form.
Consider \(\sqrt{8}\). You can simplify it because 8 can be factored into 4 and 2, where 4 is a perfect square. Therefore, \(\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}\).
  • Simplifying radicals helps in operations like addition or subtraction, especially when we need to combine like terms.
  • Breaking down the radicand into its factors often reveals components that are perfect squares, which can be easily rooted.
  • Once you have simplified, combining these radicals as like terms will become straightforward.

Mastering the simplification of radicals is crucial in algebra as it paves the way for solving more complex expressions.

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Most popular questions from this chapter

In Exercises \(133-136\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{\sqrt{20}}{8}=\frac{\sqrt{10}}{4}$$

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. To earn an A in a course, you must have a final average of at least \(90 \% .\) On the first four examinations, you have grades of \(86 \%, 88 \%, 92 \%,\) and \(84 \% .\) If the final examination counts as two grades, what must you get on the final to earn an A in the course?

Explain how to find the least common denominator for denominators of \(x^{2}-100\) and \(x^{2}-20 x+100\).

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A city commission has proposed two tax bills. The first bill requires that a homeowner pay \(\$ 1800\) plus \(3 \%\) of the assessed home value in taxes. The second bill requires taxes of \(\$ 200\) plus \(8 \%\) of the assessed home value. What price range of home assessment would make the first bill a better deal?

When graphing the solutions of an inequality, what does a parenthesis signify? What does a square bracket signify?
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