Problem 10 Simplify. $$5 \sqrt{x}+3 \sqrt... [FREE SOLUTION]

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Elementary Algebra

Laura Bracken, Ed Miller

$Math Studyset 91影视 Explanations$ Math

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Chapter 8: Problem 10

Simplify. $$5 \sqrt{x}+3 \sqrt{x}+9 \sqrt{y}-4 \sqrt{y}$$

Short Answer

Expert verified

8 \sqrt{x} + 5 \sqrt{y}

Step by step solution

Identify Like Terms

First, recognize the like terms in the expression. Like terms are terms that involve the same radicals. In this case, we have terms involving $ \sqrt{x} $ and terms involving $ \sqrt{y} $.

Combine Like Terms for $ \sqrt{x} $

Add the coefficients of the like terms involving $ \sqrt{x} $: $ 5 \sqrt{x} + 3 \sqrt{x} $. \[ 5 \sqrt{x} + 3 \sqrt{x} = (5 + 3) \sqrt{x} = 8 \sqrt{x} \]

Combine Like Terms for $ \sqrt{y} $

Add the coefficients of the like terms involving $ \sqrt{y} $: \[ 9 \sqrt{y} - 4 \sqrt{y} = (9 - 4) \sqrt{y} = 5 \sqrt{y} \]

Write the Final Simplified Expression

Combine the results from the previous steps to obtain the final simplified expression: \[ 8 \sqrt{x} + 5 \sqrt{y} \]

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Key Concepts

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

Simplifying Expressions

Simplifying expressions in algebra is a fundamental skill. It makes complex problems easier to understand and solve. When you simplify an expression, you are combining like terms to make the expression as concise as possible.

For example, in the expression given:
$5 \sqrt{x} + 3 \sqrt{x} + 9 \sqrt{y} - 4 \sqrt{y}$, we can simplify it by combining the like terms. Here's how:

Identify the terms with the same variables and radicals.
Add or subtract the coefficients of these terms.

This process helps in reducing the expression to its simplest form without changing its value. The final simplified expression from the given problem is $8 \sqrt{x} + 5 \sqrt{y}$.

Radicals

Radicals are presented as roots, such as square roots, cube roots, etc. In our expression, $\sqrt{x}$ and $\sqrt{y}$ are square roots. Simplifying radicals follows similar rules to simplifying normal algebraic expressions:

Identify the radical terms that have the same base (same variable and same index).
Combine these like terms by adding or subtracting their coefficients.

For instance, in the problem, terms like $5 \sqrt{x}$ and $3 \sqrt{x}$ are combined since they share the same radical base $\sqrt{x}$. The same goes for $9 \sqrt{y}$ and $4 \sqrt{y}$. Remember, only like radicals can be combined.

Coefficients

Coefficients are the numerical part of terms that include variables or radicals. In our expression, the coefficients are the numbers in front of $\sqrt{x}$ and $\sqrt{y}$. Here's how they work in combining like terms:

Identify the coefficients of like terms.
Add or subtract these coefficients while keeping the radical part unchanged.

For instance, $5 \sqrt{x}$ means 5 is the coefficient of $\sqrt{x}$, and $3 \sqrt{x}$ means 3 is the coefficient of $\sqrt{x}$. Their sum is $5 + 3 = 8$, resulting in $8 \sqrt{x}$. Similarly, for $9 \sqrt{y}$ and $4 \sqrt{y}$, the combined coefficient is $9 - 4 = 5$, resulting in $5 \sqrt{y}$.

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91影视

Simplify. $$5 \sqrt{x}+3 \sqrt{x}+9 \sqrt{y}-4 \sqrt{y}$$

Short Answer

Step by step solution

Identify Like Terms

Combine Like Terms for \( \sqrt{x} \)

Combine Like Terms for \( \sqrt{y} \)

Write the Final Simplified Expression

Key Concepts

Simplifying Expressions

Radicals

Coefficients

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