Problem 26 - A Graphical Approach to College Algebra

Question

If possible, simplify each radical expression. Assume that all variables represent positive real numbers. $$\sqrt{45}$$

Step-by-Step Solution

Answer

\(\sqrt{45} = 3\sqrt{5}\)

Step 1: Identify the perfect square factor

The first task is to identify a perfect square factor of 45. The number 45 can be broken down into smaller factors: \[ 45 = 9 imes 5 \] Here, 9 is a perfect square (\(3^2\) = 9).

Step 2: Break down the radical expression

Using the factorization from Step 1, rewrite the original expression by separating the perfect square from the other factor:\[ \sqrt{45} = \sqrt{9 imes 5} \]

Step 3: Apply the square root to the perfect square

Use the property of square roots that says \( \sqrt{a imes b} = \sqrt{a} imes \sqrt{b} \). Here, we take the square root of the perfect square:\[ \sqrt{45} = \sqrt{9} imes \sqrt{5} \]Knowing that \(\sqrt{9} = 3\), you can further simplify it: \[ \sqrt{45} = 3 \times \sqrt{5} \]

Step 4: Simplify the expression if needed

The expression \(3 \times \sqrt{5}\) is already in its simplest form because 5 is a prime number and cannot be simplified under the square root any further.

Key Concepts

Perfect Square Factor

A perfect square factor is a number that can be expressed as the square of an integer. These numbers are fundamental when simplifying radical expressions. In our example, we need to simplify \( \sqrt{45} \). The process begins by identifying perfect square factors. We factorize 45 to find its divisors:

• 45 = 9 × 5
• Here, 9 is a perfect square because it equals \(3^2\).

Recognizing perfect square factors simplifies the process of removing them from under the square root. This means you only keep smaller, non-perfect square numbers under the radical sign.

Square Root

The square root function allows us to find a number that, when multiplied by itself, yields the original number. In simple terms, it's the opposite of squaring a number. Every number has two square roots: positive and negative, but we typically use the positive square root when simplifying radicals.

• The square root of a perfect square like 9 is a whole number: \( \sqrt{9} = 3 \).
• For non-perfect squares, such as 5, the square root remains in the radical form: \( \sqrt{5} \).

In radical simplification exercises, understanding when a square root can be simplified to a whole number is crucial to making the expression simple.

Radical Expression Simplification

The goal of simplifying a radical expression is to express it in its most reduced form. This often involves factoring out perfect square factors. Here's how we simplify \(\sqrt{45}\) using the principles discussed:

• Recognize that 45 can be expressed as 9 × 5.
• Re-write the expression to separate the perfect square: \(\sqrt{45} = \sqrt{9 \times 5}\).
• Apply the property of radicals: \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \).
• Simplify further using \(\sqrt{9} = 3\): \(\sqrt{45} = 3 \times \sqrt{5}\).

The resulting expression \(3 \times \sqrt{5}\) is simplified as much as possible because you can't simplify \(\sqrt{5}\) any further. This process highlights the importance of recognizing perfect squares and using properties of square roots effectively.