Problem 16 Simplify. $$7 \sqrt{x}+4 \sqrt... [FREE SOLUTION]

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Elementary Algebra

Laura Bracken, Ed Miller

$Math Studyset 91影视 Explanations$ Math

1 Edition

Chapter 8: Problem 16

Simplify. $$7 \sqrt{x}+4 \sqrt{2 y}-9 \sqrt{x}+6 \sqrt{3 y}$$

Short Answer

Expert verified

-2 \sqrt{x} + 4 \sqrt{2y} + 6 \sqrt{3y}

Step by step solution

- Group Like Terms

Identify and group like terms in the expression. In this case, like terms have the same radical part: - Like terms with $\sqrt{x}$ are $7 \sqrt{x}$ and $-9 \sqrt{x}$.- Like terms with $\sqrt{2y}$ are $4 \sqrt{2y}$ and $6 \sqrt{3y}$. However, $6 \sqrt{3y}$ is not a like term with $4 \sqrt{2y}$ because their radical parts are different.

- Combine Like Terms

Combine the coefficients of like terms to simplify the expression:- For $\sqrt{x}$ terms: $7 \sqrt{x} - 9 \sqrt{x} = -2 \sqrt{x} $.- Other terms do not have like counterparts, so they remain the same.

- Write the Simplified Expression

Combine the results to write the simplified expression: $-2 \sqrt{x} + 4 \sqrt{2y} + 6 \sqrt{3y} $.

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Key Concepts

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

Like Terms

In mathematical expressions, like terms are terms that have the same variable and the same power. When simplifying algebraic expressions, it's crucial to identify these like terms. For radical expressions, like terms have the same radical part. For instance, in the exercise given, the like terms are those containing $\textstyle \sqrt{x}$, such as $7 \sqrt{x}$ and $-9 \sqrt{x}$. Recognizing these terms is the first step in simplifying the expression correctly. If terms do not have the same radical part, like $4 \sqrt{2y}$ and $6 \sqrt{3y}$, they cannot be combined.

Coefficients

Coefficients are the numerical parts of terms that multiply the variable or radical part. In our exercise, the coefficients are the numbers in front of the radicals, such as 7, 4, -9, and 6. When simplifying expressions with like terms, we specifically look at combining these coefficients. For example, with the like terms $\textstyle \sqrt{x}$, we combine 7 and -9. The subtraction $\textstyle 7 - 9 = -2$ shows that the new coefficient for $\textstyle \sqrt{x}$ is -2.

Radical Part

The radical part of a term involves the square root or other root operations. This part of the term determines whether two terms are like terms or not. In the provided exercise, the radical parts include $\sqrt{x}$, $\sqrt{2y}$, and $\sqrt{3y}$. You can only combine terms if their radical parts are identical. For example, you can combine $7 \sqrt{x}$ and $-9 \sqrt{x}$ because they both have the radical part $\sqrt{x}$. However, you cannot combine $4 \sqrt{2y}$ and $6 \sqrt{3y}$ as their radical parts, $\sqrt{2y}$ and $\sqrt{3y}$, are different.

Combine Terms

The final step in simplifying radical expressions is to combine like terms. After identifying the like terms and their coefficients, you use simple arithmetic operations to combine them. In our example, we combine $\textstyle 7 \sqrt{x}$ and $-9 \sqrt{x}$ by subtracting the coefficients: $(7 - 9) \sqrt{x} = -2 \sqrt{x}$. The terms $\textstyle 4 \sqrt{2y}$ and $\textstyle 6 \sqrt{3y}$ have no like counterparts in the expression, so they stay the same. Therefore, the simplified expression becomes $\textstyle -2 \sqrt{x} + 4 \sqrt{2y} + 6 \sqrt{3y}$.

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91影视

Simplify. $$7 \sqrt{x}+4 \sqrt{2 y}-9 \sqrt{x}+6 \sqrt{3 y}$$

Short Answer

Step by step solution

- Group Like Terms

- Combine Like Terms

- Write the Simplified Expression

Key Concepts

Like Terms

Coefficients

Radical Part

Combine Terms

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