Problem 41 - College Algebra with Corequisite Support

Question

For the following exercises, simplify each expression. \(3 \sqrt{a b^{2}}-b \sqrt{a}\)

Step-by-Step Solution

Answer

The simplified expression is \(2b\sqrt{a}\).

Step 1: Identify the Common Term

In the expression \(3 \sqrt{a b^{2}} - b \sqrt{a}\), note the presence of \(\sqrt{a}\) in both terms. This can be factored out for simplification.

Step 2: Factor Out the Common Square Root

Factor \(\sqrt{a}\) out of the expression: \(\sqrt{a} (3 \sqrt{b^{2}} - b)\). This simplifies part of the expression by bringing the square root of \(a\) outside.

Step 3: Simplify Inside the Parentheses

Inside the parentheses, recognize that \(\sqrt{b^{2}} = b\). Thus the expression becomes: \(\sqrt{a} (3b - b)\).

Step 4: Simplify the Expression

Combine the terms inside the parentheses: \(3b - b = 2b\). Therefore, the expression is reduced to \(2b\sqrt{a}\).

Key Concepts

Factoring Expressions

Factoring expressions is like unraveling a mystery hidden in algebraic terms. Imagine having several pieces of a puzzle that fit together in a particular way. In algebra, you look for common elements or factors among different terms. For instance, in the expression \(3 \sqrt{a b^{2}} - b \sqrt{a}\), we see a shared component: \(\sqrt{a}\). By factoring this out, we can simplify the expression to a clearer, more concise form.

Here's how you can approach factoring:

• Identify common factors: Look for terms that appear in each part of the expression. In our example, \(\sqrt{a}\) is present in both terms.
• Factor out the common term: Take these common factors outside the parentheses. This helps to simplify the math inside the parentheses.

With practice, factoring becomes a powerful tool in making complex algebraic expressions more manageable.

Square Roots

Square roots are one of the fundamental concepts in mathematics, often used to find the original number that was squared. Here, we need to understand how they interact with other numbers, especially in algebraic expressions.

Consider \(\sqrt{b^2}\). The square root of \(b^2\) is just \(b\), because squaring \(b\) and then square rooting it returns you to the original number. Square roots can also be factored when they are a component of a common term, as we did with \(\sqrt{a}\) in the expression. By applying the property that \(\sqrt{x^2} = x\), you can simplify complex expressions efficiently.

Here are some tips to handle square roots:

• Whenever you see \(\sqrt{x^2}\), simplify it to \(x\).
• Use square roots to simplify terms inside expressions, especially when factoring.

Understanding square roots makes dealing with quadratic expressions and other algebra topics easier.

Combining Like Terms

In algebra, combining like terms is comparable to cleaning up clutter. Often, expressions contain terms that can be simplified. Like terms are terms that have the same variable raised to the same power. When you combine them, you simplify the expression.

In the expression \(3b - b\), both terms are like terms because they contain \(b\). Combining them simplifies the expression to \(2b\). This method is crucial because it reduces the complexity of mathematical problems and helps in reaching a simplified solution faster.

Key pointers for combining like terms:

• Identify terms with the same variables exponents. These are your like terms.
• Add or subtract coefficients of these terms to simplify the expression.

Mastering the technique of combining like terms is crucial for efficient problem solving and clarity in algebra.