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Chapter 0: Problem 34

Add or subtract terms whenever possible. $$8 \sqrt{5}+11 \sqrt{5}$$

Short Answer

Expert verified
The result of \(8 \sqrt{5} + 11 \sqrt{5}\) is \(19 \sqrt{5}\).

Step by step solution

01

Identify like terms

The like terms are '8\(\sqrt{5}\)' and '11\(\sqrt{5}\)'. They're termed like terms because they both contain the same square root, \(\sqrt{5}\).
02

Combine like terms

The rule to combine like terms is similar to basic arithmetic. Just add or subtract the coefficients (the numbers in front of the square roots) while keeping the like term, \(\sqrt{5}\), the same. So here, \(8 \sqrt{5} + 11 \sqrt{5} = 19 \sqrt{5}\).

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Key Concepts

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

Radical Expressions
When we talk about radical expressions, we refer to any mathematical expression containing a root symbol. The most common type of radical expression is a square root, but there are others like cube roots or fourth roots. The key to understanding radical expressions is to recognize that they are just another form of numbers, and can be manipulated in similar ways to integers or fractions.

Dealing with radical expressions often requires simplifying them by combining like terms and potentially rationalizing denominators. In our exercise, dealing with radical expressions involves recognizing terms that can be combined because they include the same square root, such as \(8 \sqrt{5}\) and \(11 \sqrt{5}\).
Square Roots
The term square root refers to a number that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5, because 5 times 5 equals 25. When you see the symbol \(\sqrt{ }\), it represents the square root of the number inside the radical.

Understanding square roots is crucial when working with radical expressions. It's especially important to remember that square roots can have both positive and negative answers, though the principal or primary root is usually the positive one. In algebra, working with square roots often means simplifying them or adding them鈥攊f they are like terms.
Like Terms
Like terms in algebra are elements that have the same variables raised to the same power. Essentially, they are the same kind of terms, and this is what allows us to combine them through addition or subtraction. In the case of radical expressions, like terms will involve the same radical component.

For instance, the like terms in our exercise are \(8 \sqrt{5}\) and \(11 \sqrt{5}\) because they share the same \(\sqrt{5}\). This commonality makes it possible to add or subtract their numerical coefficients while keeping the radical part intact. Recognizing like terms is a vital skill when simplifying algebraic expressions.
Algebraic Addition
Algebraic addition follows the same principle as arithmetic addition鈥攍ike terms can be combined by adding their coefficients. In algebra, this process involves more than just numbers; it includes variables, exponents, and radicals as well.

To successfully perform algebraic addition, you must first identify like terms. Once identified, you can add or subtract the coefficients while maintaining the common variable or radical factor. In our example \(8 \sqrt{5} + 11 \sqrt{5} = 19 \sqrt{5}\), we add the coefficients of the like terms which both contain the square root of 5. The desired result is a simplified expression where all like terms are combined.

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