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Chapter 8: Problem 4

In each of the following, add or subtract the radical expressions. \(9 \sqrt{5}-\sqrt{5}\)

Short Answer

Expert verified
8 \sqrt{5}\

Step by step solution

01

- Identify Like Radicals

Notice that both terms have the same radical part: \(\sqrt{5}\). This means they are like radicals and can be combined.
02

- Combine the Coefficients

Combine the coefficients of the like radicals. \(9 \sqrt{5}\) and \( - \sqrt{5}\) have coefficients 9 and -1 respectively. So, compute: \(9 - 1 = 8.\)
03

- Write the Result

Attach the common radical part to the combined coefficient. The result is: \(8 \sqrt{5}\)

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Key Concepts

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

like radicals
Like radicals are similar to like terms in algebra. They have the same radical part, which is the part under the square root symbol. For example, in the exercise, both terms are paired with \( \sqrt{5} \). If they have the same number under the root, you can work with them directly. This is because the radical part ensures these terms share a common characteristic, making them like radicals.
Think of it like sorting apples and oranges. You can add or subtract apples from apples, but not apples from oranges. Similarly, you can only combine like radicals.

Here are some key points to remember:
  • Check the number under the radical sign.
  • If they鈥檙e the same, you have like radicals.
  • If they are not, you can't directly combine them.
combining coefficients
Once you've identified like radicals, the next step is to combine their coefficients. The coefficient is the number that stands in front of the radical sign. For instance, in \( 9 \:math\sqrt{5} \), 9 is the coefficient, and in \( - \math\sqrt{5} \), the coefficient is -1.

Here鈥檚 how you combine coefficients:
  • Keep the radical part the same.
  • Add or subtract only the coefficients.
  • In the exercise: \( 9 - 1 = 8 \).
After combining the coefficients, you then apply the result to the common radical part. So \( (9 \math\sqrt{5} - 1 \math\sqrt{5} = 8 \math\sqrt{5}) \). Always double-check your arithmetic to avoid mistakes.
addition and subtraction of radicals
Adding and subtracting radicals follows similar rules to combining like terms in regular algebra. If you have like radicals, you can add or subtract them just like you would add or subtract numerical coefficients.

Steps for addition and subtraction of radicals:
  • Identify like radicals (e.g., \( \sqrt{5} \) and \( \sqrt{5} \)).
  • Combine their coefficients (e.g., \( 9 - 1 = 8 \)).
  • Attach the common radical part to the result (e.g., \( 8 \math\sqrt{5} \)).
It鈥檚 crucial to align these properly since mismatched radicals (e.g., \( \sqrt{5} \) and \( \sqrt{3} \)) cannot be combined using addition or subtraction.
The final simplified result should have just one combined coefficient with the like radical, making your solution clear and concise.

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Most popular questions from this chapter

Quickly solve these square root problems. \(\sqrt{49}\)

Solve the radical expressions by finding the positive roots. \(\sqrt[3]{125}\)

In each case, simplify the radical expressions by placing them under the same radical sign. \((\sqrt{3})(\sqrt{3})\)

Solve each of the following square root problems by rewriting the radical expression as two separate expressions. \(\sqrt{(4)(11)}\)

In each of the following, add or subtract the radical expressions. \(7 \sqrt{2}+\sqrt{2}\)
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