Problem 15 - Elementary Algebra

Question

Simplify. \((79 \sqrt{3}-43 \sqrt{3})-(9 \sqrt{3}-33 \sqrt{3})\)

Step-by-Step Solution

Answer

The simplified expression is \( 60 \sqrt{3} \).

Step 1: Combine Like Terms in the First Parentheses

In the expression \( (79 \sqrt{3} - 43 \sqrt{3}) \), the terms \( 79 \sqrt{3} \) and \( -43 \sqrt{3} \) are like terms. You can combine them by subtracting the coefficients: \( 79 - 43 = 36 \). Hence, \( 79 \sqrt{3} - 43 \sqrt{3} = 36 \sqrt{3} \).

Step 2: Combine Like Terms in the Second Parentheses

Now for the expression \( (9 \sqrt{3} - 33 \sqrt{3}) \), subtract the coefficients of \( \sqrt{3} \) which are \( 9 - 33 = -24 \). So, \( 9 \sqrt{3} - 33 \sqrt{3} = -24 \sqrt{3} \).

Step 3: Subtract the Results from Steps 1 and 2

Subtract the result from Step 2 from the result from Step 1: \(36 \sqrt{3} - (-24 \sqrt{3})\). Subtracting a negative is the same as adding, so this becomes \(36 \sqrt{3} + 24 \sqrt{3}\).

Step 4: Simplify the Final Expression

Combine the like terms from the expression in Step 3: \(36 + 24 = 60\). So, \(36 \sqrt{3} + 24 \sqrt{3} = 60 \sqrt{3} \).

Key Concepts

Understanding Like Terms

When simplifying expressions, identifying like terms is crucial. Like terms are terms that have the exact same variable parts, in this case, the square root of 3, or \(\sqrt{3}\). The only difference between these terms is their coefficients:

• For example, in the expression \(79 \sqrt{3} - 43 \sqrt{3}\), both terms have the same \(\sqrt{3}\) component.
• Thus, they are like terms, and you can combine them by performing operations on their coefficients.

To combine them, you subtract or add just the coefficients, leaving the variable part ( e.g., \(\sqrt{3}\) intact. In our example, subtracting the coefficients gives 36, as seen in step 1 of the solution.

Mastering Subtraction in Expressions

Subtraction can sometimes be confusing, especially when dealing with coefficients of like terms. Here, you need to deal not only with subtracting numbers but understanding how these mathematical operations translate to expressions:

• For example, when you handle \(9 \sqrt{3} - 33 \sqrt{3}\), it's essential to focus on the coefficients, just like we did in the first parentheses.
• This subtraction of coefficients directly influences the simplified expression.

Remember that subtracting a negative is an equivalent of addition, turning things tricky at step 3 when combining results. \(36 \sqrt{3} - (-24 \sqrt{3})\) becomes an addition process. Always keep in mind how subtraction can switch to addition when managing negative numbers.

The Role of Square Roots

Square roots, like \(\sqrt{3}\), serve as the constant part of our terms in these expressions. Unlike the coefficients, which change due to addition or subtraction, the square root remains unchanged:

• They remain constant, which allows you to focus just on the numerical coefficients during calculations.
• In an expression, it represents a repeated factor within each component term.

In simplifying such expressions, the key to moving forward is to recognize these unchanged parts, concentrating on altering coefficients as opposed to the root itself. This understanding simplifies expressions like \(60 \sqrt{3}\), ensuring clarity and efficiency in mathematical operations.