Problem 60 - Algebra 2

Question

Simplify each radical expression. \(\sqrt{72}\)

Step-by-Step Solution

Answer

The expression \(\sqrt{72}\) simplifies to \(6\sqrt{2}.\)

Step 1: Factor the Number Inside the Radical

First, identify the prime factors of 72. The number 72 can be factored into 2 (as it is even) multiple times: \[72 = 2 \times 36 = 2 \times 6 \times 6 = 2 \times 2 \times 3 \times 6 = 2 \times 2 \times 3 \times 2 \times 3 = 2^3 \times 3^2\].

Step 2: Separate Perfect Squares from the Radical

Now, identify the perfect square(s) among the factors. From the previous factorization:\[ 2^3 \times 3^2 = (2^2) \times 2 \times (3^2). \] Here, \(2^2\) and \(3^2\) are perfect squares.

Step 3: Simplify the Radical by Removing Perfect Squares

Take the square root of the perfect squares outside of the radical. \[\sqrt{72} = \sqrt{2^3 \times 3^2} = \sqrt{(2^2) \times 2 \times (3^2)} = \sqrt{2^2} \times \sqrt{3^2} \times \sqrt{2} = 2 \times 3 \times \sqrt{2}.\] Thus, the expression simplifies to:\[6\sqrt{2}.\]

Key Concepts

Prime Factorization

Prime factorization is a method used to express a number as a product of its prime numbers. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. To simplify a radical expression, like \(\sqrt{72}\), we need to find its prime factors. For 72, the process begins by identifying that it is an even number, and thus divisible by 2, the smallest prime number:

• Divide 72 by 2 to get 36.
• Then, divide 36 by 2 to get 18.
• Continue dividing by 2 to get 9, which is not even, so switch to the next smallest prime, 3.
• Divide 9 by 3 to get 3, and finally, divide by 3 to reach 1.

This step-by-step process produces the prime factorization of 72 as \(72 = 2^3 \times 3^2\). Knowing the prime factors allows us to simplify radicals effectively.

Perfect Squares

Perfect squares are numbers that are the square of an integer. They play a critical role in simplifying radicals since they can be "taken out" of the radical, simplifying the expression. In our example, after finding the prime factors of 72, we organize them to identify perfect squares:

• From \(2^3 \times 3^2\), we can rearrange this as \((2^2) \times 2 \times (3^2)\).
• In this setup, \(2^2\) and \(3^2\) are perfect squares because their square roots are integers (2 and 3, respectively).

Recognizing these perfect squares allows us to simplify the radical, taking these factors outside the square root in the next steps, leading to a simpler expression.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. In our exercise, we deal with simplifying \(\sqrt{72}\). Simplification involves reducing the expression to its simplest form.The process is made clear once we have the prime factorization and identify the perfect squares. We then handle the simplification by:

• Taking the square root of the perfect squares \(\sqrt{2^2}\) and \(\sqrt{3^2}\), which results in 2 and 3 respectively.
• This pulls them out of the radical, leaving \(\sqrt{2}\) still under the square root.

So, the simplified expression becomes \(2 \times 3 \times \sqrt{2} = 6\sqrt{2}\). This simplification reduces complex radical expressions into more manageable forms, which is often the goal when dealing with mathematical problems involving radicals.