Problem 176 - Elementary Algebra

Question

In the following exercises, simplify. $$ 11 \sqrt{2 r s}-9 \sqrt{2 r s}+3 \sqrt{2 r s} $$

Step-by-Step Solution

Answer

5 \sqrt{2rs}\

Step 1: Identify like terms

Notice that all the terms in the expression share the same square root component, \( \sqrt{2rs} \). Therefore, they are like terms and can be combined.

Step 2: Combine coefficients

Add the coefficients of the like terms together: \(11 - 9 + 3\). This results in \(5\).

Step 3: Simplify the expression

Multiply the combined coefficient by the common square root component: \(5 \sqrt{2rs}\).

Key Concepts

Combining Like Terms

In algebra, simplifying expressions often involves combining like terms. Like terms are terms that have the same variable part. In our example, each term includes the square root component \( \sqrt{2rs} \). Since they share this component, they can be combined into a single term.
For instance:

• 11 \( \sqrt{2rs} \)
• -9 \( \sqrt{2rs} \)
• 3 \( \sqrt{2rs} \)

are all like terms.
Combining them involves adding or subtracting their coefficients, which are the numerical parts of the terms. This process simplifies the expression and makes it easier to work with.

Coefficients

A coefficient is the numerical part of any term in an algebraic expression or equation. In the expression from our example:

• 11 \( \sqrt{2rs} \)
• -9 \( \sqrt{2rs} \)
• 3 \( \sqrt{2rs} \)

the coefficients are 11, -9, and 3 respectively. When simplifying the expression, sum up these coefficients:
\[ 11 + (-9) + 3 = 5 \]
Thus, the combined coefficient is 5. This result multiplies the common square root component, making the simplified expression 5 \( \sqrt{2rs} \).

Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 4 is 2 because 2 \( \times \) 2 equals 4. In our example, we deal with the square roots \( \sqrt{2rs} \).
This expression signifies the square root of the product of 2, r, and s. Key properties of square roots include:

• Non-negative values - they are always greater than or equal to zero.
• \