Problem 22 - Intermediate Algebra

Question

Change each radical to simplest radical form. \(\sqrt{48}\)

Step-by-Step Solution

Answer

\(\sqrt{48} = 4\sqrt{3}\).

Step 1: Identify the Factors

First, identify the factors of the number under the square root. For \(\sqrt{48}\), the factors of 48 that are perfect squares are 16 and 3, because \(16 \times 3 = 48\).

Step 2: Rewrite the Square Root Using the Factors

Express \(\sqrt{48}\) as the product of square roots of these factors: \(\sqrt{48} = \sqrt{16 \times 3}\).

Step 3: Simplify the Square Roots

Simplify the expression by taking the square root of the perfect square factor. \(\sqrt{16} = 4\), so \(\sqrt{48} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3}\).

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics, especially when working with simplifying numbers. The square root of a number is a value that, when multiplied by itself, gives the original number. This is often represented by the radical symbol \( \sqrt{} \).
For example, the square root of 25 is 5, because \( 5 \times 5 = 25 \). Not all numbers have a neat, whole number as their square root. Numbers like 48 require further simplification. We'll explore how this works.
It's essential to understand that square roots can be both positive and negative, but in practice, we typically refer to the principal (positive) square root.

Perfect Squares

Perfect squares are numbers that are the square of an integer. This means there is a whole number which, when multiplied by itself, results in the given number. Recognizing perfect squares is crucial for simplifying square roots.
Some of the smallest perfect squares include:

• 1 (since \(1 \times 1 = 1\))
• 4 (since \(2 \times 2 = 4\))
• 9 (since \(3 \times 3 = 9\))
• 16 (since \(4 \times 4 = 16\))
• 25 (since \(5 \times 5 = 25\))

By identifying perfect square factors within numbers like 48, it becomes easier to simplify the square root. In our example, 48 can be expressed as \(16 \times 3\), where 16 is a perfect square. This allows us to take the square root of 16 directly.

Factorization

Factorization is the process of breaking down a number into its prime components, which can include perfect squares, if applicable. Factorizing numbers is a logical way to simplify square roots and fully understand the structure of any number.
To factorize 48, you could break it down into its prime factors: 48 is \(2 \times 2 \times 2 \times 2 \times 3\). From here, you can notice that \(2^4\) (which is 16, a perfect square) and 3 are its factors.
Identifying the perfect square factor (16 in our example) simplifies the operation of taking the square root. So, \(\sqrt{48}\) becomes \(\sqrt{16 \times 3}\). Knowing that \(\sqrt{16} = 4\), we simplify the expression further to \( 4\sqrt{3} \), making it the simplest radical form.