Problem 19 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ \frac{\sqrt{a^{2} b}}{\sqrt{b}} $$

Step-by-Step Solution

Answer

The simplified expression is \( a \).

Step 1: Simplifying the Radical

To simplify the expression \( \frac{\sqrt{a^{2} b}}{\sqrt{b}} \), start by simplifying the numerator. The square root of \( a^{2} b \) can be broken into \( \sqrt{a^2} \times \sqrt{b} \). Since \( \sqrt{a^2} = a \), this simplifies to \( a \sqrt{b} \).

Step 2: Cancel Common Terms

Now that you have \( \frac{a \sqrt{b}}{\sqrt{b}} \), you can cancel the \( \sqrt{b} \) in the numerator and the denominator. This leaves you with \( a \).

Step 3: Final Step: Simplified Expression

The expression is fully simplified to \( a \). Therefore, the simplified form of the given expression is \( a \).

Key Concepts

Radical Expressions

Radical expressions involve expressions containing a root, often a square root or cube root. In this exercise, we deal with a square root, which is denoted by the radical symbol \( \sqrt{} \). To simplify radical expressions, we look for opportunities to break down the terms under the radical. By recognizing perfect squares or perfect cubes, simplification becomes possible.

For example, in the expression \( \sqrt{a^2 b} \), \( a^2 \) is a perfect square. This lets us separately simplify it as \( \sqrt{a^2} \) which equals \( a \). The remaining term \( \sqrt{b} \) cannot be simplified further without additional information on \( b \), so it remains under the radical. Identifying these patterns helps simplify complex radical expressions into their easiest form.

Numerator and Denominator

In any expression written as a fraction, the top part is called the numerator, and the bottom part is the denominator. In algebra, simplifying an expression often involves handling both the numerator and the denominator wisely to get a simpler form.

For the given expression \( \frac{\sqrt{a^2 b}}{\sqrt{b}} \), \( \sqrt{a^2 b} \) is the numerator while \( \sqrt{b} \) is the denominator. Initial simplification of distinct elements in the numerator is important. By identifying simplifiable components such as perfect squares, one can quickly and effectively reduce the complexity of an expression.

Subsequent adjustment of the denominator then aligns the components so common terms can be easily seen and cancelled if possible. Understanding the roles and interactions of numerators and denominators is key when working with algebraic fractions.

Canceling Terms

Canceling terms is an essential algebraic technique used to simplify expressions. This involves reducing the expression by eliminating identical terms present in both the numerator and the denominator. However, it is crucial to ensure these terms are common factors.

In the expression \( \frac{a \sqrt{b}}{\sqrt{b}} \), we notice \( \sqrt{b} \) appears in both parts. Therefore, by cancelling \( \sqrt{b} \), we essentially divide both the numerator and denominator by \( \sqrt{b} \), simplifying the entire fraction to \( a \).

Ensure that only terms that are factors, not just components of sums or differences, are cancelled. Canceling is what reduces your expression fully, making it simpler and sometimes exposing the core value or structure hidden within the larger expression. By following this method, expressions become more understandable and manageable.