Problem 21 - Intermediate Algebra

Question

Change each radical to simplest radical form. \(\sqrt{27}\)

Step-by-Step Solution

Answer

The simplest radical form of \( \sqrt{27} \) is \( 3\sqrt{3} \).

Step 1: Identify the prime factors

First, we need to find the prime factorization of the number under the radical, which is 27. The prime factors of 27 are 3, 3, and 3, since \( 27 = 3 \times 3 \times 3 \).

Step 2: Group the prime factors

Next, we look for pairs of the same factors since the square root involves finding pairs. We can pair two of the threes together: \( (3 \times 3) \times 3 \).

Step 3: Simplify the square root

For every pair of the same number inside the square root, one of those numbers can come outside of the square root. From our pair \( (3 \times 3) \), we can take one 3 out of the square root. The expression becomes \( 3\sqrt{3} \).

Step 4: Write the simplest radical form

Now, the simplest radical form of \( \sqrt{27} \) is \( 3\sqrt{3} \), as no further simplification can be made.

Key Concepts

Prime Factorization

When you're dealing with the simplest radical form, the first step is often finding the prime factorization. But what exactly does that mean? Prime factorization is like breaking down a number into building blocks. These building blocks are called prime numbers, which are numbers only divisible by 1 and themselves. For example, the number 27 can be broken down into prime numbers like this:

• First, divide 27 by the smallest prime number, which is 3.
• You get 9 from the first division because 27 divided by 3 equals 9.
• Keep going! Divide the 9 by 3 again, the smallest prime number that fits.
• This leaves you with 3.

The end result is 3 multiplied by itself three times, or written as: \(27 = 3 \times 3 \times 3\). Now, you're set for simplifying further!
Understanding prime factorization gives you the foundation needed for square root simplification, because you see clearly which numbers can pair up for the next steps.

Square Root Simplification

Square root simplification is all about expression neatness. Once you have your prime factors lined up, you can make your square root simpler by pulling out number pairs. Here's how it looks:

• If you've got \(3 \times 3\) under the square root, it becomes a single '3' outside the square root.
• That leftover '3' from \(3 \times 3 \times 3\)? It stays inside as \(\sqrt{3}\) because it doesn't have a pair.

So, the square root expression \(\sqrt{27}\), with its simplified form, becomes \(3\sqrt{3}\).
Simplifying square roots by finding pairs reduces the complexity of radical expressions. It translates our rich forest of numbers into much simpler paths.

Radical Expressions

Dealing with radical expressions often feels like uncovering hidden simplicity in complex numbers. Once you've simplified using prime factorization and square root simplification, you're left with a neat radical expression.

• The expression \(3\sqrt{3}\) tells a clear story. It's made of a coefficient (the 3 outside) and a radical (\(\sqrt{3}\)).
• The coefficient multiplies the radical, helping to scale and adjust the value of your entire expression.

In practice, you'll often find yourself simplifying radical expressions to better handle them in algebra or to compare sizes and values more easily.
This helps in solving equations or evaluating mathematical problems with increased clarity and understanding.