Problem 41 - College Algebra

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: Factor out the Common Term

Both terms in the expression contain the term \( \sqrt{a} \). Begin by factoring \( \sqrt{a} \) out of the expression. The expression becomes: \[ 3b \sqrt{a} - b \sqrt{a} \].

Step 2: Simplify Using Distribution

Now that we have factored \( \sqrt{a} \), distribute the common factor.Rewrite the expression as: \[ (3b - b) \sqrt{a} \].

Step 3: Simplify the Coefficients

Calculate the coefficient by simplifying \(3b - b\).This yields: \[ 2b \].

Step 4: Write the Final Simplified Expression

Combine the simplified coefficient with the factored square root term. The final expression is: \[ 2b \sqrt{a} \].

Key Concepts

Factoring Common Terms

Factoring common terms is a crucial step in simplifying algebraic expressions. When simplifying an expression, always look for terms that share a common factor. In our exercise, both terms have the common factor \( \sqrt{a} \). By factoring \( \sqrt{a} \) out of the expression, we essentially "group" the terms together to reveal a simpler form.

Here's how it works:

• Identify common terms in the expression.
• "Factor out" the shared term to simplify the expression.

By factoring out \( \sqrt{a} \), the original expression \( 3 \sqrt{a b^{2}} - b \sqrt{a} \) is transformed into \( 3b \sqrt{a} - b \sqrt{a} \), significantly simplifying the process of dealing with the remaining coefficients and terms.

Distribution in Algebra

Distribution in algebra involves applying a factor to multiple terms inside a set of parentheses. This is also known as the distributive property. Once you've factored out a common term, you often need to apply distribution to further simplify the expression.

In our example, after factoring out \( \sqrt{a} \), we're left with \( (3b - b) \). This step of distribution allows you to see the structure of the expression more clearly:

• Rewrite the expression inside the parentheses: \( (3b - b) \).
• Apply distribution, focusing on simplifying what's inside the parentheses.

By handling the coefficients correctly, you see that \( 3b - b \) becomes \( 2b \), which further helps streamline the entire expression.

Square Roots in Algebra

Square roots can often make algebra problems look intimidating. However, understanding how they work can simplify expressions effectively. In algebra, a square root such as \( \sqrt{a} \) is a number that, when multiplied by itself, gives the original value of \( a \).

Here are some important things to keep in mind:

• Square roots can be treated like variables when simplifying expressions.
• Look for opportunities to factor out square root terms, simplifying the expression.

In our step-by-step solution, \( \sqrt{a} \) was successfully factored out, leading to the final simplified version of the expression: \( 2b \sqrt{a} \). Remember, handling square roots with confidence can greatly assist in simplifying complex algebraic problems.