Problem 76 - Algebra 1

Question

Simplify the radical expression. $$ \sqrt{24} $$

Step-by-Step Solution

Answer

The simplified form of the radical expression \(\sqrt{24}\) is \(2\sqrt{6}\).

Step 1: Prime Factorization of 24

Start the simplification process by breaking down the number under the square root, 24, into its prime factors. The prime factorization of 24 is \(2 \times 2 \times 2 \times 3\), or \(2^3 \times 3^1\).

Step 2: Pair up the factors

For a square root, every pair of similar factors can be moved out of the root. Since there are three 2's, two of them can pair up and move outside of the root. The one unpaired 2 and 3 remain under the root.

Step 3: Simplifying the radical expression

By moving the pair of 2’s outside the root, the simplified radical expression becomes \(2\sqrt{6}\).

Key Concepts

Understanding Prime Factorization

Prime factorization is an essential math skill that involves breaking down a composite number into its basic building blocks, the prime numbers.
Here, a prime number is a number greater than 1 that can only be divided by itself and 1 without leaving a remainder.
To find the prime factorization of a number:

• Start by dividing the number by the smallest prime number, which is 2.
• If the number is divisible by 2, continue to divide by 2 until you no longer get a whole number.
• Then, move on to the next smallest prime number, which is 3, and repeat the process.
• Continue this pattern, moving to 5, 7, 11, and so forth, until the remaining quotient is a prime number.

Taking 24 as an example, we divide by 2 three times: \(24 \div 2 = 12,\) \(12 \div 2 = 6,\) and \(6 \div 2 = 3\)
The process stops when we reach 3, a prime number.
Thus, the prime factorization of 24 is \(2^3 \times 3^1\), written as a product of powers of prime numbers.

Decoding Square Roots

The square root is a mathematical symbol that finds a number which, when multiplied by itself, results in the original number inside the square root sign.
It is denoted by the radical symbol \(\sqrt{}\).
When simplifying square roots:

• Look for perfect square factors in the number under the square root.
• Remember that any factor that pairs up completely can be "brought out" from under the square root.

For example, when simplifying \(\sqrt{24}\), first identify its prime factors \(2^3 \times 3^1\).
Among these, \(2^3\) has one pair of 2s.
The pair can "escape" the square root as a single factor, \(2\), leaving behind the unpaired \(2\) and \(3\) inside the square root.
This results in the simplified form \(2\sqrt{6}\), because \(6 = 2 \times 3\).
Understanding square roots paves the way for simplifying many more complex radical expressions with ease.

Simplifying Radical Expressions

Simplifying radical expressions is a process that helps express roots in their simplest form.
This involves factoring out perfect squares from under the radical symbol to streamline the expression.
Follow these steps to simplify a radical expression:

• Perform prime factorization on the number inside the radical.
• Organize the prime factors and identify any pairs.
• For every pair, take one factor out of the square root.
• Multiply any remaining prime factors inside the radical together.

Consider simplifying \(\sqrt{24}\):

• The prime factorization yields \(2^3 \times 3^1\).
• Pair up the 2's: one pair of \(2\) moves outside of the square root.
• The simplified form, then, is \(2\sqrt{6}\).

This process clarifies how some numbers can break out of their radical enclosure, thereby simplifying the expression.
It's a valuable tool in solving more advanced mathematical problems and makes it easier to perform further calculations.