Problem 4 - Elementary and Intermediate Algebra

Question

Fill in the blanks. To _____ \(\sqrt{24}\) means to write it as \(2 \sqrt{6}.\)

Step-by-Step Solution

Answer

Simplifying \(\sqrt{24}\) to \(2 \sqrt{6}\) is called simplifying.

Step 1: Understand the Meaning

The exercise requires us to identify the process of rewriting the square root of a number by factoring it into simpler components. This is a common technique used to simplify square roots.

Step 2: Factor the Number

We begin by factoring 24 into its prime factors: 24 can be factored as \(24 = 2^3 \times 3\).

Step 3: Simplify Using Square Roots

Since we have \(\sqrt{24} = \sqrt{2^3 \times 3}\), we can separate the factors to simplify: \(\sqrt{24} = \sqrt{2^2 \times 2 \times 3}\).

Step 4: Apply Square Root Rules

Apply the property \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\): \(\sqrt{2^2 \times 2 \times 3} = \sqrt{2^2} \times \sqrt{2 \times 3}\).

Step 5: Calculate Perfect Squares

Since the square root of a perfect square is the number itself, we find \(\sqrt{2^2} = 2\). Now write it as: \(2 \times \sqrt{6}\).

Step 6: Write the Conclusion

Therefore, the process of rewriting \(\sqrt{24}\) as \(2 \sqrt{6}\) is known as simplifying.

Key Concepts

Prime Factorization

Prime factorization is a technique used to break down a number into its prime number constituents. Prime numbers are those greater than 1, which only have two divisors: 1 and themselves. These numbers include things like 2, 3, 5, 7, 11, and so forth.
In the context of simplifying square roots, knowing how to factor a number into its prime components is invaluable. Let's take the example of 24:

• Divide 24 by the smallest prime number, which is 2, giving you 12.
• Divide 12 by 2 again to get 6.
• Divide 6 by 2 to obtain 3.
• Notice, 3 is a prime number itself, so the factorization stops here.

So, the prime factorization of 24 is expressed as: \( 2^3 \times 3 \). Knowing this primes the pump for the next step in simplifying square roots.

Properties of Square Roots

Square roots have unique properties that allow us to simplify expressions involving them. The property pertinent to our task is \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). This property lets us split square roots across their multiplicative components.
If you know the prime factorization of a number, you can apply this property effectively. Back to our example, \( \sqrt{24} \) can be rewritten as \( \sqrt{2^3 \times 3} \). Breaking it down further, this is \( \sqrt{2^2 \times 2 \times 3} \). Notice how the perfect square, \( 2^2 \), is isolated here.
Applying the square root property, we get:

• \( \sqrt{2^2} \times \sqrt{2 \times 3} \)

This separation primes us to calculate individual square roots, which further simplifies the whole expression.

Perfect Squares

Perfect squares are numbers that can be expressed as some integer squared. For instance, 1, 4, 9, 16, 25, etc., are all perfect squares. The significance of these numbers in simplifying square roots is that the square root of a perfect square is always an integer.
In our simplification process, we found \( \sqrt{2^2} \) in the expression \( \sqrt{2^2 \times 2 \times 3} \). The square root of \( 2^2 \) is 2 since \( 2^2 = 4 \), and thus \( \sqrt{4} = 2 \). Recognizing and extracting such perfect squares from under a square root sign is crucial.
After calculating \( \sqrt{2^2} = 2 \), the expression \( \sqrt{24} \) then becomes \( 2 \times \sqrt{6} \), which is the simplified form. Leveraging perfect squares in this manner helps streamline the simplification.