Problem 6 - Prealgebra

Question

Simplify each expression by taking as much out from under the radical as possible. You may assume that all variables represent positive numbers $$\sqrt{48}$$

Step-by-Step Solution

Answer

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

Step 1: Identify Perfect Squares

To simplify \(\sqrt{48}\), we first look for perfect square factors of 48. A perfect square is a number like 4, 9, 16, 25, etc., that has an integer as its square root. Here, the largest perfect square factor of 48 is 16, since \(16 \times 3 = 48\).

Step 2: Rewrite the Radicand

Express 48 as a product of 16 and 3: \(\sqrt{48} = \sqrt{16 \times 3}\).

Step 3: Apply the Square Root Property

Use the property of radicals, \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). Therefore, \(\sqrt{48} = \sqrt{16} \times \sqrt{3}\).

Step 4: Simplify the Square Root

Calculate the square root of the perfect square: \(\sqrt{16} = 4\). So, \(\sqrt{48} = 4 \times \sqrt{3}\).

Step 5: Final Simplification

Write down the simplified expression: \(\sqrt{48} = 4\sqrt{3}\). This is the simplest form of the expression.

Key Concepts

Perfect Squares

When dealing with radicals, identifying perfect squares is the first step to simplifying them. A perfect square is a number that can be expressed as the square of an integer. For example, 1, 4, 9, 16, 25, 36, and so forth. These numbers are crucial because their square roots are integers, which makes them simpler to work with.
In problems like simplifying \(\sqrt{48}\), finding the largest perfect square that divides into the radicand (the number under the radical sign) gives us a cleaner path to simplification. Here, for 48, our perfect square is 16 because \(16 \times 3 = 48\), and \(16\) is itself a perfect square (\(4^2\)). Identifying such factors helps in breaking down the radicand to a simpler form, paving the way for further simplification using other properties.

Square Root Property

The square root property is a fundamental concept used in simplifying radicals. The property states that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\).
This property allows us to split up a product inside a square root into separate square roots, which can make it easier to further simplify each part.
For example, with \(\sqrt{48}\), since 48 can be rewritten as \(16 \times 3\) (where 16 is a perfect square), we use the square root property as follows:

• \(\sqrt{48} = \sqrt{16 \times 3}\)
• \(\sqrt{48} = \sqrt{16} \times \sqrt{3}\)

This breakdown is useful as it isolates the perfect square part (\(\sqrt{16}\)) from the rest, which we can then simplify separately.

Integer Factorization

Integer factorization complements the process of simplifying radicals. It involves breaking down a number into its basic factors, typically focusing on prime numbers and perfect squares.
To simplify \(\sqrt{48}\), we need to determine the factors of 48. The factors include 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. Among these, 16 is a perfect square, which is pivotal to simplifying the expression.
Here's how you can think about factorization in this context:

• Start by breaking down the radicand: 48.
• Decompose it into 16 and 3 (since \(16 \times 3 = 48\)).
• Notice 16 is \(4^2\), hence it is easy to work with under a square root.

Overall, integer factorization is a crucial strategy in simplifying radicals, setting the stage for using perfect squares and the square root property.