Problem 10 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(2\sqrt{6}\).

Step 1: Prime Factorization of 24

Breakdown 24 into its prime factors. This can be done by dividing 24 by the smallest prime number, which is 2. The prime factorization of 24 is 2 x 2 x 2 x 3 or, in exponent form, \(2^3 \cdot 3^1\).

Step 2: Pairing and Simplifying

In a square root, pairs of the same factor simplify to just the factor itself outside the square root. Here, \(2^3\) can be rewritten as \(2 \cdot 2^1\) (or just 2). So, one pair of 2 can be taken out of the square root leaving \(2^1 \cdot 3^1\) (or just 6) inside the square root.

Step 3: Final Simplified Expression

Outside the square root we have the number that was taken out, which is 2, and inside the square root we still have 6. This gives us the simplified expression \(2\sqrt{6}\).

Key Concepts

Prime Factorization

Understanding prime factorization is a foundational concept in mathematics, particularly when simplifying square roots. Prime factorization involves breaking down a composite number into the smallest prime numbers that, when multiplied together, give the original number.
Let's take the number 24 as an example. To find its prime factors, begin dividing by the smallest prime number, which is 2, repeatedly until you can't anymore.

• First, divide 24 by 2 to get 12.
• Then, divide 12 by 2 to get 6.
• Next, divide 6 by 2 to get 3.
• Since 3 is itself a prime number, we stop here.

Through this process, we find that 24 can be expressed as a product of prime numbers: \(2^3 \times 3^1\). Recognizing these building blocks is crucial for the next steps in simplification.

Square Roots

A square root represents a value that, when multiplied by itself, gives the original number. Simplifying square roots is about expressing the root in its simplest form by removing perfect squares.
The square root symbol \( \sqrt{} \) typically asks us to find a number whose square equals the number inside the symbol.
For example, with \( \sqrt{24} \), the goal is to express this in a simpler form. After performing prime factorization, we aim to identify any pairs of numbers since pairs of the same number under the square root can be simplified.
Being able to simplify square roots efficiently is invaluable, helping to make calculations easier and results more decipherable.

Pairing Factors

Pairing factors is a technique used to simplify square roots after breaking a number down into its prime factors. It involves identifying pairs of identical numbers under the square root and removing them. This concept relies on the property that the square of a number can move outside the square root.
Take the prime factors of 24, which are \(2^3 \times 3^1\). Here, \(2^3\) indicates three 2s being multiplied together. You can pair up two of those 2s. Each full pair of numbers underneath a square root symbol becomes a single number outside of it.

• A pair like \(2 \times 2\) comes out of the square root as a single 2, simplifying \(\sqrt{2^3 \times 3}\) to \(2\sqrt{6}\).

The result reflects an essential simplification in mathematical expressions, providing a streamlined and more comprehensible form.