Problem 41 - College Algebra

Question

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 Common Factors

Look at both terms of the expression: the first term is \(3 \sqrt{ab^2}\) and the second term is \(b \sqrt{a}\). Notice that both terms contain \(\sqrt{a}\) as a factor.

Step 2: Rewrite Each Term

Rewrite both terms to clearly identify common factors. For \(3 \sqrt{ab^2}\), rewrite it as \(3b \sqrt{a}\). The second term, \(b \sqrt{a}\), already appears in this form.

Step 3: Combine Like Terms

Having rewritten both terms as \(3b \sqrt{a}\) and \(b \sqrt{a}\), observe that they have the common factor \(b \sqrt{a}\). Simply subtract the coefficients (3 and 1): \((3b - b) \sqrt{a} = 2b \sqrt{a}\).

Key Concepts

Common Factors

When simplifying expressions, identifying common factors is essential. It helps in reducing expressions to simpler forms. In the expression given, we have two terms:

• The first term is \(3 \sqrt{ab^2}\)
• The second term is \(b \sqrt{a}\)

Common factors are elements or variables that appear in both terms. In this case, both terms contain \(\sqrt{a}\) as a common factor. Recognizing these factors allows us to rewrite the terms in a way that makes simplification easier. By focusing on these common elements, we can combine the terms effectively in the later steps of the process.

Combining Like Terms

Like terms in algebraic expressions are those that have the same variables raised to the same powers. When combining like terms, you simply add or subtract the coefficients of these terms while keeping the common variables intact.

• In our example, after rewriting the first term as \(3b \sqrt{a}\), we observe both terms: \(3b \sqrt{a}\) and \(b \sqrt{a}\).
• These share the common factor \(b \sqrt{a}\), making them like terms.

To simplify, we subtract the coefficients, 3 and 1 (implicitly present in \(b \sqrt{a}\)), resulting in \((3b - b) \sqrt{a} = 2b \sqrt{a}\). This step is crucial as it consolidates the expression to its simplest form.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. Simplifying these expressions involves ensuring that the expression is in its simplest form, with no further reductions possible. This includes factoring within the radical and outside simplifying terms or factors.

• In the example we worked through, both \(\sqrt{ab^2}\) and \(\sqrt{a}\) are radical expressions.
• Identifying that \(b^2\) can be simplified under the radical to \(b\) helps to rewrite and simplify the expression.

By comprehending the properties of radicals and working with them, we achieve an expression like \(2b \sqrt{a}\), with no radicals left unfactored or unsimplified. Understanding these principles lays a solid foundation for manipulating and simplifying other radical expressions effectively.