Problem 18 - College Algebra

Question

Simplify each expression. $$\sqrt{192}$$

Step-by-Step Solution

Answer

The simplified form of \( \sqrt{192} \) is \( 8 \sqrt{3} \).

Step 1: Prime Factorization

To simplify \( \sqrt{192} \), we start with finding the prime factorization of 192. We divide by the smallest prime number 2 until we can't anymore: 192 ÷ 2 = 96 96 ÷ 2 = 48 48 ÷ 2 = 24 24 ÷ 2 = 12 12 ÷ 2 = 6 6 ÷ 2 = 3 Finally, 3 is a prime number. So, 192 = 2^6 \times 3.

Step 2: Pair the Prime Factors

In square roots, we can pair the factors. In the factorization of 192 which is \( 2^6 \times 3 \), we can form pairs of 2: So, \( \sqrt{192} = \sqrt{(2^6) \times 3} = \sqrt{((2^3) imes (2^3)) \times 3} \).

Step 3: Simplify Using Paired Factors

A pair of factors (\( a^2 \)) under a square root can be moved outside the square root as a single factor (\( a \)). So for "pairs of 2s": \( \sqrt{(2^3) \times (2^3) \times 3} = 2^3 \times \sqrt{3} \).

Step 4: Calculate the Simplified Expression

Now calculate \( 2^3 \):\( 2^3 = 8 \).Thus, \( \sqrt{192} = 8 \sqrt{3} \).

Key Concepts

Prime Factorization

Prime factorization is a method of breaking down a number into its basic building blocks: prime numbers. Prime numbers are numbers that can only be divided by 1 and themselves, like 2, 3, 5, and 7. To find the prime factorization of a number, you start dividing the number by the smallest prime number, which is 2. You continue dividing by 2 until the number is no longer divisible by 2, then move to the next prime number, which is 3, and so on.

• For the number 192:
• First, divide by 2: 192 ÷ 2 = 96.
• Keep dividing by 2: 96 ÷ 2 = 48, 48 ÷ 2 = 24, 24 ÷ 2 = 12, 12 ÷ 2 = 6, and 6 ÷ 2 = 3.
• Because 3 is a prime number, you can stop here.

The prime factorization of 192 is expressed as the product of prime numbers: \(192 = 2^6 \times 3\).

Pairing Factors

When simplifying roots, it helps to think about pairing factors, especially when dealing with square roots. When calculating the square root, you're essentially asking: "What number multiplied by itself gives me this original number?" By arranging the prime factors into pairs, you can easily see which ones can "come out" of the square root.

• From the prime factorization \(2^6 \times 3\), focus on the 2s:
• There are six 2's, or \(2^6\), which means we can create pairs:
• You can pair them as \( (2^3) \times (2^3) \) under the square root.

In mathematical form, you are preparing to move out pairs: \( \sqrt{(2^6) \times 3} \rightarrow \sqrt{(2^3) \times (2^3) \times 3} \).

Square Roots Simplification

The process of simplifying square roots involves reducing the root to its simplest form. After pairing factors of the prime factorization, you can simplify the expression.The rule here is: any pair of prime factors inside the square root can be taken outside as a single factor.

• From our example: \(\sqrt{(2^3) \times (2^3) \times 3}\).
• Each \(2^3\) inside the square root becomes a 2 outside because \(\sqrt{(2^3) \cdot (2^3)} = 2^3\).
• As there's only one 3, it stays inside the root: \(\sqrt{3}\).

Thus, the entire expression simplifies to \(2^3 \times \sqrt{3}\). Since \(2^3 = 8\), the final simplified form is \(8\sqrt{3}\). With practice, you'll find that simplifying square roots is all about recognizing pairs and understanding when they can exit the square root sign as whole numbers.