Problem 155 - Elementary Algebra

Question

In the following exercises, simplify. $$ 5 \sqrt{c}+2 \sqrt{c} $$

Step-by-Step Solution

Answer

The simplified expression is \(7 \sqrt{c}\).

Step 1: Understand the Expression

Identify that the expression contains like terms involving square roots. The terms are \(5 \sqrt{c}\) and \(2 \sqrt{c}\).

Step 2: Combine Like Terms

Since both terms have \( \sqrt{c} \), these can be combined by simply adding the coefficients. Thus, add 5 and 2 to get \(5 + 2 = 7\).

Step 3: Write the Simplified Expression

Multiply the resultant coefficient by the square root term to get \(7 \sqrt{c}\).

Key Concepts

Like Terms

In algebra, combining like terms is crucial for simplifying expressions. Like terms are terms that have identical variable parts. For example, in the given exercise, the terms \(5 \sqrt{c}\) and \(2 \sqrt{c}\) are like terms because both contain the variable \sqrt{c}\.
When combining like terms:

• Identify the terms that have the same variable and exponent.
• Add or subtract the coefficients (the numerical parts) of these terms.
• Keep the variable part unchanged.

So, in the example, we identify \(5 \sqrt{c}\) and \(2 \sqrt{c}\) as like terms and simply add their coefficients (5 + 2) to get 7. This results in the simplified expression \(7 \sqrt{c}\).

Coefficients

A coefficient is the numerical part of a term that is multiplied by the variable part. In the expression \(5 \sqrt{c}\), the number 5 is called the coefficient, and \sqrt{c}\ is the variable part.
Key points about coefficients:

• They can be positive or negative numbers.
• They represent how many times the variable part is multiplied.
• When terms have the same variable part, their coefficients can be added or subtracted.

For example, in the given exercise, we have coefficients 5 and 2 associated with the variable \sqrt{c}\. By adding these coefficients, we get a new coefficient, which is 7. The expression then becomes \(7 \sqrt{c}\).

Square Roots

Square roots are a special kind of mathematical operation that asks for the number which, when multiplied by itself, gives the original number. In algebra, the square root is often represented with the radical symbol \sqrt{}\. For example, \sqrt{25} = 5\ because \5 \times 5 = 25\.
In the context of algebraic expressions:

• Square roots with the same radicand (the number or expression inside the square root) are considered like terms and can be combined.
• We treat the variable part inside the square root as a single entity.

In the exercise, \sqrt{c}\ is the common variable part in both terms \(5 \sqrt{c}\) and \(2 \sqrt{c}\). By combining the coefficients of these like terms, we simplify the expression to \(7 \sqrt{c}\).