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Chapter 10: Problem 8

Add or subtract as indicated. $$8 \sqrt{17}-5 \sqrt{19}-6 \sqrt{17}+4 \sqrt{19}$$

Short Answer

Expert verified
The simplified expression is \(2 \sqrt{17} - \sqrt{19}\)

Step by step solution

01

Identify the like terms

In the given expression, \(8 \sqrt{17}\) and \(-6 \sqrt{17}\) are like terms. Similarly, \(-5 \sqrt{19}\) and \(4 \sqrt{19}\) are also like terms since they both contain square root of 19.
02

Combine the like terms

The like terms can be combined by adding or subtracting them. For the square root of 17, the expression will be \(8 \sqrt{17} - 6 \sqrt{17}\) which simplifies to \(2 \sqrt{17}\). For the square root of 19, the expression will be \(-5 \sqrt{19} + 4 \sqrt{19}\) which simplifies to \(-1 \sqrt{19}\) or simply \(-\sqrt{19}\).
03

Write down the final answer

After adding or subtracting the like terms, we get the simplified expression as \(2 \sqrt{17} - \sqrt{19}\). This is the final answer.

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Key Concepts

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

Understanding Like Terms
When dealing with expressions in algebra, it's important to recognize *like terms*. Like terms are terms that have the same variables raised to the same power. In the context of radicals, like terms have the same radical part. Let's take a closer look at an example:
  • In the expression \(8\sqrt{17} - 5\sqrt{19} - 6\sqrt{17} + 4\sqrt{19}\), the terms \(8\sqrt{17}\) and \(-6\sqrt{17}\) are like terms because they both involve \(\sqrt{17}\).
  • Similarly, \(-5\sqrt{19}\) and \(4\sqrt{19}\) are like terms because they both involve \(\sqrt{19}\).
Like terms can be easily added or subtracted because they share the same base element, like the radical here, allowing us to combine their coefficients. Recognizing and combining like terms accurately is crucial to simplifying expressions.
Simplifying Expressions
Simplifying expressions involves reducing them to their simplest form. This process makes calculations easier and expressions more manageable. When dealing with like terms, you simplify by:
  • Identifying like terms. This includes terms with the same radical or the same variable.
  • Combining these terms by adding or subtracting their coefficients.
In our example, we started by simplifying \(8\sqrt{17} - 6\sqrt{17}\). Since these are like terms, you can combine them by subtracting their coefficients:
  • \(8 - 6 = 2\), leading to the simplified term \(2\sqrt{17}\).
Similarly, with \(-5\sqrt{19} + 4\sqrt{19}\):
  • \(-5 + 4 = -1\), simplifying to \(-1\sqrt{19}\) or \(-\sqrt{19}\).
Properly simplifying expressions by combining like terms ensures that you arrive at the most straightforward result.
Radicals Explained
Radicals are symbols used to denote root values, commonly the square root, \(\sqrt{}\). They are crucial in algebra and various mathematical applications. A radical expression contains a radical symbol with a number or expression beneath it, for example, \(\sqrt{17}\).
  • The number beneath the radical sign is called the radicand. In \(\sqrt{17}\), 17 is the radicand.
  • Combining radicals only works when the radicands are the same, just like with like terms.
When simplifying expressions that involve radicals:
  • Check if the radicals have the same radicand. If yes, treat them like terms.
  • Apply arithmetic operations to the coefficients of the radicals.
In our problem, radicals share the same radicand 鈥� 17 and 19 in pairs 鈥� allowing us to simplify by adding or subtracting. Understanding how radicals function and are manipulated is vital to successfully simplifying expressions in algebra.

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Most popular questions from this chapter

In Exercises \(75-92,\) rationalize each denominator. Simplify, if possible. $$\frac{2 \sqrt{x}+\sqrt{y}}{\sqrt{y}-2 \sqrt{x}}$$

In Exercises \(39-64,\) rationalize each denominator. $$\frac{10}{\sqrt[5]{16 x^{2}}}$$

In Exercises \(39-64,\) rationalize each denominator. $$\frac{5}{\sqrt[4]{x}}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equations \(\sqrt{x+4}=-5\) and \(x+4=25\) have the same solution set.

What is an extraneous solution to a radical equation?
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