Problem 59 - Algebra 2

Question

Simplify each radical expression. \(\sqrt{48}\)

Step-by-Step Solution

Answer

The simplified form of \(\sqrt{48}\) is \(4\sqrt{3}\).

Step 1: Identify Perfect Square Factors

First, we will find the prime factors of the number under the radical, which is 48. The prime factorization of 48 is: \[ 48 = 2^4 \times 3 \]Among these, \(2^4\) contains \(2^2 = 4\), which is a perfect square.

Step 2: Separate the Perfect Square Factor

Express \(\sqrt{48}\) using the perfect square factor:\[ \sqrt{48} = \sqrt{2^4 \times 3} = \sqrt{(2^2)^2 \times 3} = \sqrt{4^2 \times 3} \]

Step 3: Simplify the Radical

Using the product rule of radicals \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\), simplify:\[ \sqrt{48} = \sqrt{4^2} \times \sqrt{3} = 4 \times \sqrt{3} \]

Step 4: Final Expression

Now we have the simplified form of the radical:\[ \sqrt{48} = 4\sqrt{3} \]

Key Concepts

Perfect Square Factors

When we talk about perfect square factors, we mean numbers that are squares of integers. These numbers, when placed under a square root, dissolve into their original integer form. For example, 4 is a perfect square because it can be written as \(2^2\). Identifying perfect square factors within a larger number under a radical is a key step in simplifying radicals.

Let's consider the number 48 under the square root, \(\sqrt{48}\). We need to break down 48 into its factors and detect which of them are perfect squares. The number 48 can be factored into 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. Among these, 4 and 16 are perfect squares. However, to simplify, we'll use the largest grouping of squares, which is found from its prime factorization.

Prime Factorization

Prime factorization involves expressing a number as a product of prime numbers. This method helps us simplify radicals effectively as it makes locating perfect square factors easier.

To factor the number 48 into primes, we repeatedly divide by the smallest prime number until we are left with nothing but prime numbers:

• 48 divided by 2 is 24
• 24 divided by 2 is 12
• 12 divided by 2 is 6
• 6 divided by 2 is 3

Since 3 is a prime number, we stop here, thus the prime factorization of 48 is \(2^4 \times 3\). This decomposition plays a crucial role in recognizing perfect square factors such as \(2^2 = 4\), which simplifies the process of extracting values from under the square root.

Product Rule of Radicals

The product rule of radicals is a fundamental principle that allows us to break down and simplify expressions under a square root sign. The rule states:

• \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)

This rule signifies that the square root of a product is the product of the square roots. Applying this helps us split the square root of a larger number into more manageable components, especially when dealing with perfect square factors.

In our example, after conducting prime factorization and identifying \(2^4\) as \(4^2\), we can rewrite \(\sqrt{48}\) using the product rule as follows:

• \(\sqrt{48} = \sqrt{4^2} \times \sqrt{3}\)
• Simplifies to \(4 \times \sqrt{3}\)

By leveraging the product rule of radicals, complex radicals can be broken down into simpler forms, facilitating easy comprehension and calculation.