Problem 38 - College Algebra with Modeling and Visualization

Question

Simplify the radical expression by factoring out the largest perfect nth power. Assume that all variables are positive. $$ \sqrt{72} $$

Step-by-Step Solution

Answer

\(6 \sqrt{2}\)

Step 1: Identify a Factorization

Start by factoring the number 72 into its prime factors. The factors of 72 are: \(72 = 2^3 \times 3^2\). This helps us to break the radical into smaller, manageable parts.

Step 2: Break Down the Square Root

Use the factored form from Step 1 to write the entire radical expression in terms of square roots of those factors: \( \sqrt{72} = \sqrt{2^3 \times 3^2} \).

Step 3: Simplify by Pairing

Within the square root, look for pairs of the same number, as these can be brought out of the square root as single numbers. For \(2^3\), we have one pair of 2s, and \(3^2\) is a perfect square (two pairs of 3s).

Step 4: Factor Out Perfect Squares

Factor these pairs out of the radical: \( \sqrt{2^3 \times 3^2} = \sqrt{(2^2) \times 2 \times 3^2} \). Since \( \sqrt{2^2} = 2 \) and \( \sqrt{3^2} = 3 \), these both come out of the square root.

Step 5: Final Simplification

Multiply the numbers you factored out from underneath the radical: \(2 \cdot 3 = 6\). So the simplified form of the expression is \(6\sqrt{2}\).

Key Concepts

Factorization

Factorization is the process of breaking down a number into its simplest components that, when multiplied together, result in the original number. Imagine you want to understand how a large building is made. You might break it down into its bricks and beams. Similarly, when simplifying radicals, we break the number under the square root down to its prime factors. This makes the task manageable, just like dealing with individual building blocks.

For instance, in the exercise, the number 72 can be expressed as a product of its prime factors:

• The number 2, raised to the third power: \(2^3\) (which means 2 multiplied by itself three times).
• The number 3, raised to the second power: \(3^2\) (3 multiplied by itself).

Therefore, 72 equals \(2^3 \times 3^2\). Knowing how to factor numbers correctly is crucial for simplifying radicals because it lays the groundwork for extracting square roots or any other roots efficiently. Being skilled in factorization not only helps in math but is useful for solving many real-world problems.

Square Roots

Square roots can be visualized as the opposite operation of squaring a number. When you square a number, you multiply it by itself. Conversely, when you square root a number, you are trying to find a number which, when multiplied by itself, equals the original number under the square root.

Consider the square root of 72: \(\sqrt{72}\). With factorization, we have broken 72 into \(2^3 \times 3^2\). We can express \(\sqrt{72}\) as \(\sqrt{2^3 \times 3^2}\). The focus is to identify perfect squares within this expression, as perfect squares come out of the square root neatly.

• From \(2^3\), the pair \(2^2\) can be taken out as a single 2.
• From \(3^2\), both 3's come out since \(3^2\) is a perfect square.

This enables us to simplify \(\sqrt{72}\) by pairing and extracting these squares, showing how square roots simplify handling large or complex numbers.

Radical Expressions

Radical expressions involve roots - such as square roots, cube roots, and so on - and they are a blend of numbers and symbols that represent these roots. When you simplify a radical expression, your goal is to rewrite it in the simplest form, often described as a 'cleaner' expression.

Using our ongoing example, the original radical expression is \(\sqrt{72}\). After factorization and applying the concept of square roots, we simplified it to \(6\sqrt{2}\). Here's a quick guide to simplifying radical expressions:

• Factor the number: Break down the number under the radical sign into its prime factors.
• Identify pairs: Look for pairs of numbers. These can be "paired" and moved outside the radical sign.
• Simplify: Multiply the numbers coming out of the radical. Leave any unpaired factors inside the radical.

This final step results in a more concise and often more useful form of the radical. Understanding radical expressions is essential for solving more advanced problems in algebra and calculus.