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Chapter 9: Problem 168

In the following exercises, simplify. $$ 7 \sqrt{5}+\sqrt{5}-8 \sqrt{10} $$

Short Answer

Expert verified
The simplified expression is \( 8 \sqrt{5} -8 \sqrt{10} \).

Step by step solution

01

Identify Like Terms

First, identify the like terms in the expression. Like terms are terms that have the same radical part. In this case, the like terms are those with \( \sqrt{5} \): \(7 \sqrt{5} \) and \( \sqrt{5}\). The term \( -8 \sqrt{10}\) has a different radical part, \( \sqrt{10}\), so it will not combine with the others.
02

Combine Like Terms

Combine the like terms \( 7 \sqrt{5} \) and \( \sqrt{5} \): \( 7\sqrt{5} + \sqrt{5} = 8\sqrt{5}\).
03

Rewrite Simplified Expression

Now, rewrite the expression by including the term that did not combine. The simplified expression is: \( 8 \sqrt{5} - 8 \sqrt{10} \).

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Key Concepts

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

like terms
When simplifying expressions, understanding 'like terms' is crucial. Like terms refer to terms in an expression that have the same variable raised to the same power. This applies to radical expressions as well. For example, in the expression \(7 \sqrt{5} + \sqrt{5} - 8 \sqrt{10}\), \(7 \sqrt{5}\) and \(\sqrt{5}\) are like terms because they both have a \(\sqrt{5}\) component. On the other hand, \(-8 \sqrt{10}\) is not a like term with the others because its radical part is \(\sqrt{10}\).
radical expression
A radical expression involves roots, such as square roots, cube roots, etc. The symbol \(\sqrt{}\) usually denotes the square root. Simplifying such expressions often forms part of algebra. To illustrate, the given expression \(7 \sqrt{5} + \sqrt{5} - 8 \sqrt{10}\) contains square roots. To simplify a radical expression, it's helpful to identify like terms first. Sometimes, the radicals need to be simplified themselves, like turning \(\sqrt{20}\) into \(2\sqrt{5}\). Practicing with different radical expressions can build familiarity and confidence.
combining like terms
Combining like terms is a fundamental algebraic skill. It simplifies expressions and makes them easier to work with. Here's a step-by-step method:

1. **Identify like terms**: Look for terms that have the same variable and root. For our case, \(7 \sqrt{5}\) and \(\sqrt{5}\) are like terms.
2. **Add or subtract coefficients**: Combine the coefficients (numerical parts) of the like terms. So, \(7 \sqrt{5} + \sqrt{5} = 8 \sqrt{5}\).
3. **Rewrite the expression**: Include any terms that didn't combine. The given expression simplifies to \(8\sqrt{5} - 8\sqrt{10}\).
Practicing this process can make more complex expressions easier to handle.

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