Problem 21 - Algebra for College Students

Question

For Problems \(21-74\), 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: Prime Factorization

Begin by finding the prime factorization of the number inside the square root. The number 27 can be factored into 3 multiplied by 3 multiplied by 3, or written using exponents as \(3^3\).

Step 2: Simplify Using Square Roots

Next, apply the property of square roots that allows you to separate the square root of a product into the product of separate square roots. Using \(\sqrt{27} = \sqrt{3^3}\), you can write it as \(\sqrt{3^2 \cdot 3}\).

Step 3: Extract Perfect Squares

Identify and extract perfect squares from under the square root. Since \(3^2 = 9\) and its square root is 3, you can take this 3 out of the square root. This simplifies \(\sqrt{3^2 \cdot 3}\) to \(3 \sqrt{3}\).

Key Concepts

Prime Factorization

Prime factorization is the process of expressing a number as the product of its prime factors. Prime numbers are those greater than 1 that have no divisors other than 1 and themselves. In the case of simplifying radicals, prime factorization helps us break down complex numbers into simpler components, which is particularly useful for handling square roots.
For example, consider the number 27. To find its prime factorization, divide 27 by the smallest prime number, which is 3, because 27 is not divisible evenly by 2. Thus, we have:

• 27 divided by 3 equals 9
• 9 divided by 3 again equals 3
• Finally, 3 divided by 3 equals 1

This means 27 can be expressed as the product of three 3s, or more succinctly, as a power in the form of \(3^3\). The benefit of finding the prime factorization becomes evident when simplifying square roots, as it helps identify perfect squares.

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. Square roots are often represented by the radical symbol \(\sqrt{}\). When simplifying expressions involving square roots, one powerful property is that the square root of a product can be separated into the product of square roots.Using the exercise with \(\sqrt{27}\) as an example, the process starts by rewriting 27 using its prime factors derived from the prime factorization step: \(\sqrt{3^3}\). Next, decompose \(3^3\) into a perfect square to facilitate further simplification into the form \(\sqrt{3^2 \cdot 3}\). Using the property \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\), we can separate the square root into \(\sqrt{3^2} \cdot \sqrt{3}\).
This lays the groundwork for extracting perfect squares and simplifying the overall radical expression.

Perfect Squares

Perfect squares are numbers that are the squares of integers. Recognizing perfect squares is crucial when simplifying square roots because it allows for certain terms to be easily extracted from under the radical sign.In the example of simplifying \(\sqrt{27}\), we identified that \(3^2 = 9\) is a perfect square. Its square root is simply 3. This is key to simplifying expressions because once \(9\) is identified as a perfect square in the expression \(\sqrt{3^2 \cdot 3}\), it can be extracted to become \(3\), leaving \(\sqrt{3}\) still under the radical. Hence, \(\sqrt{27}\) simplifies to \(3\sqrt{3}\).
Understanding how to recognize and handle perfect squares allows students to efficiently simplify radical expressions, ultimately making them appear less complex. Remember, simplicity often lies in recognizing patterns, like perfect squares, within the math problem.