Problem 154 - Elementary Algebra

Question

In the following exercises, simplify. $$ \sqrt{b}-6 \sqrt{b} $$

Step-by-Step Solution

Answer

-5 \(\sqrt{b}\)

Step 1: Identify Like Terms

First, recognize that both terms, \(\sqrt{b}\) and \(-6 \sqrt{b}\), are like terms since they both contain the same radical expression, \(\sqrt{b}\).

Step 2: Combine Like Terms

Combine the like terms by subtracting the coefficients of \(\sqrt{b}\). We have \(\sqrt{b} - 6 \sqrt{b}\).

Step 3: Calculate the Coefficients

Subtract the coefficients: \1 - 6 = -5\. So, \(\sqrt{b} - 6 \sqrt{b}\) simplifies to \(-5 \sqrt{b}\).

Key Concepts

Understanding Like Terms

When simplifying radical expressions, recognizing like terms is crucial. Like terms are terms that have the same variable raised to the same power or terms that target the same root. For instance, in the problem \(\text{\color{blue}}{\sqrt{b} - 6 \text{\color{blue}}{\sqrt{b}}}\), both terms contain the radical \(\text{\color{blue}}{\sqrt{b}}\). Since they involve the same radical expression, they are considered like terms. This simplification falls into the same category as when you combine \text{\color{green}}{5x} and \text{\color{green}}{3x} in algebra because \”\text{\color{green}}{5x \text{\color{green}}{ + 3x}}\” involves the variable \textcolor{green}{x}. Like terms make it easier to perform arithmetic operations and simplify the equation. This foundational knowledge will help in solving more complex expressions.

What Are Coefficients?

In any algebraic expression, coefficients are the numerical parts that stand before variables or radicals. For example, in the term \(\text{\color{blue}}{-6 \text{\color{blue}}{\sqrt{b}}}\), \(\textcolor{blue}{-6}\) is the coefficient, and \(\text{\color{blue}}{\sqrt{b}}\) is the radical component. Identifying coefficients is important for combining like terms. Coefficients are also seen in regular algebra; \text{\color{green}}{5x} has a \textcolor{green}{5} as its coefficient. In our example, the terms \(\text{\color{blue}}{\sqrt{b}}\) and \(\text{\color{blue}}{-6 \text{\color{blue}}{\sqrt{b}}}\) have implicit coefficients of \(\text{\color{green}}{1}\) and \(\text{\color{blue}}{-6}\), respectively. Understanding coefficient values helps you simplify equations more efficiently by allowing you to add, subtract, or manipulate the numerical parts.

Combining Like Terms

Combining like terms simplifies expressions by consolidating them into a single term. Consider the exercise \(\textcolor{blue}{\text{\color{blue}}{\sqrt{b}} - 6 \text{\color{blue}}{\sqrt{b}}}\). Both terms have \(\text{\color{green}}{\sqrt{b}}\) so we can combine the coefficients. As a first step, identify the coefficients: \(\text{\color{green}}{1} + (-6) = -5}\). Therefore, \(\text{\color{blue}}{\sqrt{b}} - 6 \text {\color{blue}}{\sqrt{b}} = \text{\color{blue}}{-5 \text {\color{blue}}{\sqrt{b}}}\). This step makes your final answer simpler and more readable. Remember, always focus on like terms to avoid mistakes. Practicing this foundational method reinforces understanding and helps in solving more complex math problems with ease.