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Chapter 9: Problem 165

In the following exercises, simplify. $$ 3 \sqrt{3}-8 \sqrt{3}+7 \sqrt{5} $$

Short Answer

Expert verified
-5 \sqrt{3} + 7 \sqrt{5}

Step by step solution

01

Identify like terms

Identify terms that have the same radical part. In this expression, the terms are: \(3 \sqrt{3}\), \(-8 \sqrt{3}\), and \(7 \sqrt{5}\).
02

Combine like terms

Add or subtract the coefficients of the like terms. For \(3 \sqrt{3}\) and \(-8 \sqrt{3}\), combine them: \[ 3 \sqrt{3} - 8 \sqrt{3} = (3 - 8) \sqrt{3} = -5 \sqrt{3} \]
03

Include the remaining term

After combining like terms, make sure to include any remaining terms that do not have the same radical part. In this case, add the \(7 \sqrt{5}\) term to the result: \[ -5 \sqrt{3} + 7 \sqrt{5} \]

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Key Concepts

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

Like Terms
In algebra, 'like terms' are terms that contain the exact same variable(s) raised to the same power(s). For simplifying purposes, look at the radical part of each term. Consider the expression given in the exercise: \[ 3 \sqrt{3} - 8 \sqrt{3} + 7 \sqrt{5} \] Here, \( 3 \sqrt{3} \) and \( -8 \sqrt{3} \) are like terms because they share the same radical part, \( \sqrt{3} \).
However, \( 7 \sqrt{5} \) is different because it has \( \sqrt{5} \), not \( \sqrt{3} \). Identifying like terms is crucial because only like terms can be directly combined.
Radicals
Radicals involve the root of a number, usually indicated by the square root symbol \( \sqrt{} \). For example, \( \sqrt{3} \) and \( \sqrt{5} \) are different because they have different numbers under the root.
When simplifying radical expressions, always check to see which radicals can be combined by examining the number inside the radical symbol.

In the expression \[ 3 \sqrt{3} - 8 \sqrt{3} + 7 \sqrt{5} \], \( \sqrt{3} \) and \( \sqrt{5} \) cannot be combined with each other because they are different radicals.
Coefficients
Coefficients are the numerical parts of the terms that multiply the variables or radicals. For example, in \( 3 \sqrt{3} \), the coefficient is 3.
In \( -8 \sqrt{3} \), the coefficient is -8. And in \( 7 \sqrt{5} \), the coefficient is 7.
When combining like terms, you perform the arithmetic operations on the coefficients while keeping the radical part the same.
For instance: \[ 3 \sqrt{3} - 8 \sqrt{3} = (3 - 8) \sqrt{3} = -5 \sqrt{3} \] Here, you subtract the coefficients while retaining the \( \sqrt{3} \).
Combining Terms
Combining terms is the process of adding or subtracting like terms to simplify an expression. Always identify which terms can be combined by checking their coefficients and radicals.
From the solution steps:
  • Combine \( 3 \sqrt{3} \) and \( -8 \sqrt{3} \):
  • This gives: \((3 - 8) \sqrt{3} = -5 \sqrt{3} \).

After combining these like terms, don't forget to include any remaining terms: \[ -5 \sqrt{3} + 7 \sqrt{5} \] Since \( \sqrt{3} \) and \( \sqrt{5} \) are different radicals, they remain as separate terms in the simplified expression.

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