Problem 15 - Elementary Algebra

Question

Simplify each expression by performing the indicated operation. $$ 3 \sqrt{7 x}+2 \sqrt{7 x} $$

Step-by-Step Solution

Answer

Question: Simplify the expression: \(3\sqrt{7x} + 2\sqrt{7x}\) Answer: \(5\sqrt{7x}\)

Step 1: Identify common factors

Both terms have the same square root, which is \(\sqrt{7x}\).

Step 2: Add the coefficients

Since the square roots are the same in both terms, we can simply add the coefficients. The coefficients are 3 and 2, and $$3 + 2 = 5$$.

Step 3: Write the final expression

The simplified expression is the sum of the coefficients multiplied by the common square root: $$5\sqrt{7x}$$.

Key Concepts

Understanding Square Roots

Square roots are a fundamental concept in mathematics and are represented by the radical symbol, often written as \( \sqrt{} \). The square root of a number is a value that, when multiplied by itself, gives the original number. Understanding square roots is essential for simplifying expressions. For instance, in our example, both terms have \( \sqrt{7x} \), the square root factor that stays constant in the expression.

• This means we can treat \( \sqrt{7x} \) as a single unit or factor in our calculations.
• The square root indicates a multiplication operation snugly lurking beneath a seemingly simple expression—try to think of it as a kind of stealthy factor in your math problem.

Recognizing these square root factors can significantly simplify the process of solving or simplifying mathematical expressions.

The Role of Coefficients

Coefficients are the numbers placed in front of variables or terms in an algebraic expression. They tell us how many times a term is counted. In the expression \(3 \sqrt{7x} + 2 \sqrt{7x}\), the numbers 3 and 2 are the coefficients of the term \( \sqrt{7x} \).

• The coefficients guide us in combining like terms—more so when those terms contain the same radical or variable component.
• In our exercise, once we recognize that \( \sqrt{7x} \) is common, we only need to focus on adding these coefficients.

Adding the coefficients simplifies the given expression, demonstrating the importance of correctly identifying and operating on these numerical values.

Addition of Like Terms

Adding like terms is a crucial step in simplifying algebraic expressions. Like terms are terms with the same variable part raised to the same power, in our case, \( \sqrt{7x} \). By identifying and grouping like terms, you streamline the process of simplifying expressions.
Understanding that \(3 \sqrt{7x}\) and \(2 \sqrt{7x}\) are like terms allows us to directly add their coefficients (3 and 2), making the simplification straightforward.
As soon as the coefficients are added (resulting in 5), they multiply the shared radical part \( \sqrt{7x} \).
Thus, recognizing and efficiently combining like terms can greatly reduce complexity and make algebraic manipulation easier and more intuitive.