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Chapter 0: Problem 34

In Exercises \(33-44,\) add or subtract terms whenever possible. $$8 \sqrt{5}+11 \sqrt{5}$$

Short Answer

Expert verified
The solution is \(19 \sqrt{5}\).

Step by step solution

01

Identify Like Terms

The given terms are \(8 \sqrt{5}\) and \(11 \sqrt{5}\). Like terms are terms that have the same variable and exponent. Here, both terms can be thought of as being multiplied by \(\sqrt{5}\), so they are like terms and can be added.
02

Add Like Terms

When adding like terms, keep the common part (i.e., the \(\sqrt{5}\)) the same and add the coefficients (i.e., the numbers being multiplied by the \(\sqrt{5}\)). In this case, \(8 + 11 = 19\). Therefore, \(8 \sqrt{5} + 11 \sqrt{5} = 19 \sqrt{5}\).

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Key Concepts

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

Add Like Terms
Understanding how to add like terms is crucial in simplifying expressions. Like terms are terms that have identical variable parts, which means both the variable and its exponent must be the same. When you're tasked with adding or subtracting like terms, the procedure is straightforward:
  • Identify the Coefficients: Look at the numbers right in front of the variable part. In our example, those numbers are 8 and 11.
  • Add the Coefficients: Simply add these numbers together, as you would with any numbers: 8 + 11 gives you 19.
  • Keep the Variables: The variable part, here represented by \(\sqrt{5}\), remains unchanged when combining like terms.

By following these steps, the terms 8\( \sqrt{5} \) and 11\( \sqrt{5} \) combine to form 19\( \sqrt{5} \). This method ensures you're correctly simplifying the expression without altering its value.
Identifying Like Terms
To successfully add or subtract terms, you need to identify like terms first. Like terms in an expression have the same variable raised to the same power. For the given expression, both terms, 8\( \sqrt{5} \) and 11\( \sqrt{5} \), are multiplied by the same radical, \(\sqrt{5}\).This characteristic means they are like terms, which makes them compatible for addition.
  • Check the Variables: Ensure the expressions have the same variable and power.
  • Verify Matching Parts: Even with radicals, they must be identical (e.g., \(\sqrt{5}\) here) for terms to be considered 'like'.

Recognizing like terms enables you to combine them efficiently, allowing you to simplify complex expressions with ease.
Radical Expressions
A radical expression is any expression that includes a root, such as the square root, cube root, etc. In this case, \(\sqrt{5}\) is part of both original terms in the problem.When dealing with radical expressions, it is key to treat the radical part as you would a variable when identifying like terms.
  • Understand the Radical: Recognize that \sqrt{}\ is a mathematical operation that asks "what number times itself equals the number under the root?"
  • Handling Radicals in Operations: When adding radicals, the root must be identical for the terms to be combined.

This understanding allows you to handle operations such as addition with radicals, as demonstrated in the exercise with terms like 8\( \sqrt{5} \) and 11\( \sqrt{5} \). By treating them similarly to like terms, you simplify using the same rules you've learned for variables.

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Most popular questions from this chapter

Perform the indicated operations. $$\frac{1}{x^{n}-1}-\frac{1}{x^{n}+1}-\frac{1}{x^{2 n}-1}$$

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. A truck can be rented from Basic Rental for \(\$ 50\) per day plus \(\$ 0.20\) per mile. Continental charges \(\$ 20\) per day plus \(\$ 0.50\) per mile to rent the same truck. How many miles must be driven in a day to make the rental cost for Basic Rental a better deal than Continental's?

Perform the indicated operations. Simplify the result, if possible. $$\left(\frac{2 x+3}{x+1} \cdot \frac{x^{2}+4 x-5}{2 x^{2}+x-3}\right)-\frac{2}{x+2}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation \((2 x-3)^{2}=25\) is equivalent to \(2 x-3=5\).
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