Problem 13 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ \frac{\sqrt{45}}{\sqrt{5}} $$

Step-by-Step Solution

Answer

The simplified expression is 3.

Step 1: Simplify Under the Square Root

The expression is \[\frac{\sqrt{45}}{\sqrt{5}}.\]Simplify \(\sqrt{45}\) first. Since 45 can be factored as \(9 \times 5\), we have:\[\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3 \times \sqrt{5}.\]

Step 2: Simplify the Fraction

The expression has now become:\[\frac{3 \times \sqrt{5}}{\sqrt{5}}.\]Since \(\sqrt{5}\) appears in both the numerator and the denominator, we can cancel it out, giving us:\[3.\]

Key Concepts

Square Roots

Square roots are an essential part of mathematics, allowing us to determine the original number that was squared to produce a given value. When dealing with square roots, it's vital to remember that they have specific properties that can simplify calculations. For example:\[\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}.\]
This means that we can separate the numbers inside the square root into their factors to simplify them. In the provided exercise, \(45\) was broken down into \(9 \times 5\), and by taking the square root of \(9\), which is \(3\), the expression \(\sqrt{45}\) simplifies to \(3 \times \sqrt{5}\).
Recognizing square numbers like \(9 = 3^2\) can make simplifying radicals more efficient. Practice identifying these perfect squares to make dealing with more complex expressions easier.

Fraction Simplification

Simplifying fractions helps in reducing expressions to their simplest form, making them easier to work with or interpret. When dealing with fractions involving square roots, the process is similar to reducing any other fraction: you try to cancel out common terms from the numerator and the denominator.
For the expression \(\frac{3 \times \sqrt{5}}{\sqrt{5}}\), \(\sqrt{5}\) is present both in the numerator and the denominator. We can cancel \(\sqrt{5}\) because dividing anything by itself (except zero) equals one. Thus, the expression simplifies directly to \(3\).
It’s important to ensure terms are only canceled when the operation adheres to proper mathematical rules. Cancellations in radical expressions often simplify complex questions into their most basic form, making further calculations easier.

Mathematical Expressions

Working with mathematical expressions involves understanding symbols and operations to find a solution or simplified form. An expression like \(\frac{\sqrt{45}}{\sqrt{5}}\) involves both radicals and fractions, requiring multiple steps to simplify. Recognizing patterns, such as radical simplification and fraction reduction, is key.
Mathematical expressions can include numbers, operators, and variables, each playing a role in how the expression is interpreted and simplified. With practice, complex expressions become clearer, allowing for step-by-step reductions into their most straightforward forms.

• Identify parts of the expression like radicals or fractions.
• Apply mathematical rules like breaking square roots into factors.
• Cancel common terms sensibly.

Understanding how to approach and manipulate these expressions can significantly enhance problem-solving skills and help in tackling more advanced topics.