Simplifying Square Root Of 240 Prime Factorization Method

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When dealing with square roots, simplifying them often involves expressing the radicand (the number inside the square root) in its prime factored form. This approach allows us to identify pairs of identical factors, which can then be extracted from the square root. Let's delve into the process of using prime factorization to simplify square roots and address the specific question of simplifying √240.

Prime Factorization Explained

Prime factorization is the process of breaking down a composite number into its prime number constituents. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself (examples include 2, 3, 5, 7, 11, and so on). To find the prime factorization of a number, we repeatedly divide it by the smallest prime number that divides it evenly, continuing until we are left with only prime factors. This method is crucial for simplifying square roots and other mathematical operations.

For example, to find the prime factorization of 240, we can follow these steps:

  1. Divide 240 by the smallest prime number, 2: 240 ÷ 2 = 120
  2. Divide 120 by 2: 120 ÷ 2 = 60
  3. Divide 60 by 2: 60 ÷ 2 = 30
  4. Divide 30 by 2: 30 ÷ 2 = 15
  5. Since 15 is not divisible by 2, try the next prime number, 3: 15 ÷ 3 = 5
  6. 5 is a prime number, so we stop here.

Thus, the prime factorization of 240 is 2 × 2 × 2 × 2 × 3 × 5, often written as 2⁴ × 3 × 5. This prime factorization is the cornerstone of simplifying the square root of 240.

Simplifying Square Roots

The process of simplifying square roots using prime factorization involves rewriting the number under the square root as a product of its prime factors. We then look for pairs of identical factors. For each pair, one factor can be brought outside the square root. This simplification is based on the property that √(a²)= a, where a is a non-negative number.

Consider the square root of 240, which can be written as √240. Using the prime factorization we found earlier (2⁴ × 3 × 5), we can rewrite √240 as √(2 × 2 × 2 × 2 × 3 × 5). Now, we identify pairs of identical factors:

  • We have two pairs of 2s (2 × 2 and 2 × 2).

For each pair, one 2 can be taken out of the square root. Thus, from the two pairs of 2s, we take out 2 × 2 = 4. The remaining factors (3 and 5) stay inside the square root. This process effectively reduces the number inside the square root while expressing the original number in its simplest form.

Correctly Simplifying √240

To correctly simplify √240, we use the prime factorization 2⁴ × 3 × 5, which can be rewritten as √(2 × 2 × 2 × 2 × 3 × 5). We pair the identical factors:

  • √(2 × 2 × 2 × 2 × 3 × 5) = √(2² × 2² × 3 × 5)

For each pair of 2s, one 2 comes out of the square root:

  • √(2² × 2² × 3 × 5) = 2 × 2 × √(3 × 5)
  • = 4√15

Therefore, the simplest form of √240 is 4√15. This result is achieved by correctly identifying and extracting pairs of factors from the prime factorization of the radicand.

Common Mistakes to Avoid

When simplifying square roots using prime factorization, it's easy to make mistakes if the process is not followed meticulously. Here are some common errors to watch out for:

  1. Incomplete Prime Factorization: Failing to break down the number completely into its prime factors. This can lead to an incorrect simplification because you might miss pairs of factors.
  2. Incorrect Pairing: Misidentifying pairs of factors. It is crucial to accurately pair identical factors to extract them correctly from the square root.
  3. Arithmetic Errors: Making mistakes in the multiplication or division during prime factorization or simplification. Accuracy in each step is essential for obtaining the correct answer.
  4. Forgetting Remaining Factors: Failing to include the remaining factors inside the square root after extracting pairs. The factors that do not form pairs must stay inside the square root.
  5. Misunderstanding the Simplest Form: Thinking the expression is simplified when there are still perfect square factors inside the square root. The simplest form should have no perfect square factors remaining inside the square root.

By being aware of these common mistakes and carefully following the steps of prime factorization and simplification, you can ensure accurate results.

Practical Examples and Exercises

To solidify your understanding, let's walk through a few more examples and exercises. These examples will help you practice the steps involved in simplifying square roots using prime factorization, ensuring you grasp the concept thoroughly.

Example 1: Simplify √180

  1. Prime Factorization: Break down 180 into its prime factors.
    • 180 ÷ 2 = 90
    • 90 ÷ 2 = 45
    • 45 ÷ 3 = 15
    • 15 ÷ 3 = 5
    • So, 180 = 2 × 2 × 3 × 3 × 5
  2. Rewrite under Square Root: Rewrite √180 using its prime factors: √(2 × 2 × 3 × 3 × 5)
  3. Identify Pairs: Identify pairs of identical factors: √(2² × 3² × 5)
  4. Extract Pairs: For each pair, bring one factor out of the square root: 2 × 3 × √5
  5. Simplify: Multiply the factors outside the square root: 6√5
    • Therefore, √180 simplified is 6√5.

Example 2: Simplify √72

  1. Prime Factorization: Break down 72 into its prime factors.
    • 72 ÷ 2 = 36
    • 36 ÷ 2 = 18
    • 18 ÷ 2 = 9
    • 9 ÷ 3 = 3
    • So, 72 = 2 × 2 × 2 × 3 × 3
  2. Rewrite under Square Root: Rewrite √72 using its prime factors: √(2 × 2 × 2 × 3 × 3)
  3. Identify Pairs: Identify pairs of identical factors: √(2² × 2 × 3²)
  4. Extract Pairs: For each pair, bring one factor out of the square root: 2 × 3 × √2
  5. Simplify: Multiply the factors outside the square root: 6√2
    • Therefore, √72 simplified is 6√2.

Practice Exercises

Now, let's try some exercises to reinforce your understanding. Simplify the following square roots using prime factorization:

  1. √98
  2. √150
  3. √252

By working through these examples and exercises, you’ll become more proficient in simplifying square roots using prime factorization. This skill is valuable not only in mathematics but also in various fields that require quantitative analysis.

Conclusion

In summary, simplifying square roots using prime factorization is a fundamental technique in mathematics. By breaking down the number under the square root into its prime factors, identifying pairs, and extracting them, we can express the square root in its simplest form. Understanding and applying this method correctly not only simplifies calculations but also enhances problem-solving skills in various mathematical contexts.

Which calculation accurately uses prime factorization to write √240 in its simplest form?