Problem 36 - College Algebra Essentials

Question

$$4 \sqrt{13 x}-6 \sqrt{13 x}$$

Step-by-Step Solution

Answer

The simplified result of the given expression is \(-2 \sqrt{13 x}\)

Step 1: Identifying like terms

Here, \(4 \sqrt{13 x}\) and \(-6 \sqrt{13 x}\) are like terms because they both have the same variable base (\( \sqrt{13 x} \)).

Step 2: Applying the basic algebra rule

According to the rule that 'Like terms can be added or subtracted by adding or subtracting their coefficients', we subtract the coefficients of the like terms. Hence, \(4 \sqrt{13 x}-6 \sqrt{13 x}\) equals to \((-2) \sqrt{13 x}\)

Key Concepts

Basic Algebra

Basic algebra forms the foundation for all mathematical relations and operations involving variables and constants. One key rule of basic algebra is known as combining like terms. This rule allows for the simplification of expressions, making them easier to handle.

When we talk about like terms, we are looking for terms within an expression that have the same variable part. Even if they have different coefficients, as long as the variables and their exponents are identical, they can be combined.

In the exercise, the expression is provided as \(4 \sqrt{13 x} - 6 \sqrt{13 x}\). Each term includes the same square root, \(\sqrt{13 x}\). This makes them like terms and qualifies them to be combined simply by manipulating their coefficients.

Coefficients

In an algebraic term, the coefficient is the numerical factor that multiplies the variable part. It provides the term with its "weight" or "magnitude."

When dealing with coefficients in expressions with like terms, it is essential to remember the principle of adding or subtracting these numerical factors while keeping the variable part unchanged.

For example, in \(4 \sqrt{13 x} - 6 \sqrt{13 x}\), the coefficients are 4 and -6. To simplify, we merely perform the arithmetic operation on these coefficients: 4 - 6 which results in -2. This operation gives us the final simplified term: \(-2 \sqrt{13 x}\).

Understanding coefficients helps you manipulate and simplify algebraic expressions effectively, making complex problems more manageable.

Square Roots

Square roots are a common mathematical operation where a number or expression is multiplied by itself to return to the original value. For example, \(\sqrt{x}\) implies a number which, when squared, results in \(x\).

In the context of our expression, \(\sqrt{13 x}\) holds the position of the variable part, remaining consistent across like terms. It's crucial to observe that during operations with like terms, such as addition or subtraction, the square root component does not change; only the coefficients are manipulated.

This notion allows us to simplify expressions such as \(4 \sqrt{13 x} - 6 \sqrt{13 x}\) where the square root \(\sqrt{13 x}\) stays intact. Rather than altering this, we focus on simplifying the numerical coefficients to make operations more straightforward.