Problem 30 - College Algebra

Question

In Exercises \(27-38,\) add or subtract terms whenever possible. $$4 \sqrt{13 x}-6 \sqrt{13 x}$$

Step-by-Step Solution

Answer

-2\sqrt{13x}

Step 1: Identifying like terms

Looking at the given expression, we recognize that \(4\sqrt{13x}\) and \(-6\sqrt{13x}\) are like terms because they both have the same radical part \(\sqrt{13x}\). When we have like terms, these can be combined using the basic principles of algebra.

Step 2: Combining like terms

Now, to combine the like terms, you do the normal addition or subtraction of the coefficients of the terms, which in this case are 4 and -6. Thus, \(4\sqrt{13x} - 6\sqrt{13x}\) becomes \((4 - 6)\sqrt{13x}\), simplifying further to \(-2\sqrt{13x}\).

Key Concepts

Combining Like Terms

When working with algebraic expressions, one of the primary objectives is to simplify them as much as possible by **combining like terms**. You might wonder what "like terms" are. In simple terms, these are components within an expression that have the exact same variables raised to the same power. Consider the expression given in the exercise: \(4 \sqrt{13x} - 6 \sqrt{13x}\). The terms are identified as "like" because both terms include the same radical part \(\sqrt{13x}\). This shared radical is a key factor in identifying like terms.

• Same variables
• Same powers
• Identical radical parts (for expressions involving radicals)

Once you have identified like terms, the next step is to combine them. This is done by adding or subtracting their coefficients. In the exercise, the numbers 4 and -6 are the coefficients, and they can be combined by performing basic arithmetic: \((4 - 6)\). Doing so results in \(-2\). Therefore, the original expression simplifies to \(-2 \sqrt{13x}\).

Radicals

**Radicals** may seem daunting at first, but with practice, you'll get the hang of handling them in algebraic expressions. A radical is essentially a symbol that denotes the root of a number. In this context, understanding and manipulating radicals is crucial as they frequently occur in algebraic problems. The radical symbol \(\sqrt{}\) we encountered in the expression \(4\sqrt{13x} - 6\sqrt{13x}\) represents the square root.
Key things to remember about radicals:

• The expression under the radical symbol is called the "radicand".
• Like terms involving radicals must have identical radicands to be combined.
• Radicals can sometimes be simplified further by factoring out perfect squares.

In our expression, \(\sqrt{13x}\) is the radical part, shared by both terms, making it easier to combine using the steps outlined for like terms. Ensure you treat the radical part as a whole when considering it in your calculations.

Coefficients

In algebra, the term **coefficients** refers to the number placed in front of a variable or radical that multiplies with it. They are vital in combining like terms. For instance, in our example, \(4\sqrt{13x}\) and \(-6\sqrt{13x}\), the coefficients are 4 and -6, respectively. If you visualize the term \(\sqrt{13x}\) as a common factor, identifying the coefficients becomes much simpler.

To make the math operation more obvious, picture this: coefficients operate like simple arithmetic numbers applied to the variable or radical they accompany.

• Positive coefficients increase the term.
• Negative coefficients decrease or potentially "reverse" the direction of the term.

By thinking of coefficients as multipliers of the term they're paired with, you can easily refine and consolidate algebraic expressions. Thus, combining terms involves straightforward arithmetic. In our simplified expression, the calculations led us to \(-2 \sqrt{13x}\) when we combined 4 and -6, showing the critical role of coefficients in the process.