Problem 15 - Fundamentals of Mathematics

Question

Determine the value of each power and root. \(\sqrt{225}\)

Step-by-Step Solution

Answer

The square root of 225 is 15.

Step 1: Understand the Problem

We need to find the square root of 225, denoted by \( \sqrt{225} \). This involves finding a number which, when multiplied by itself, equals 225.

Step 2: Find the Prime Factors of 225

First, we find the prime factors of 225 to easily determine its square root. We can divide 225 by 5, which is the smallest prime number that works: \( 225 \div 5 = 45 \). Then, divide 45 by 5 again to get \( 9 \), and finally, divide 9 by 3 to get \( 3 \). Thus, the prime factors of 225 are 5, 5, 3, and 3.

Step 3: Group Prime Factors in Pairs

Once we have the prime factors, we group them into pairs to find the square root. So, we have \( (3 \times 3) \) and \( (5 \times 5) \).

Step 4: Multiply One Factor from Each Pair

Take one number from each prime factor pair and multiply them: \( 3 \times 5 = 15 \).

Step 5: Verify the Result

To ensure that 15 is the correct square root, multiply it by itself: \( 15 \times 15 = 225 \). Since this equals the original number, \( \sqrt{225} = 15 \) is verified as correct.

Key Concepts

Prime Factorization

Prime factorization is a method of breaking down a number into components called prime numbers, which are only divisible by 1 and themselves. In our example, we want to work with the number 225. To perform prime factorization, you start by dividing the number by the smallest prime number possible.

• First, we divide 225 by 5, the smallest prime factor, to get 45.
• Next, divide 45 again by 5 to yield 9.
• Finally, divide 9 by 3 to reach 3.

As a result, the prime factors of 225 are these numbers: 5, 5, 3, and 3. This step is important because it lays the groundwork for determining the square root by simplifying the original number into smaller components. Understanding prime factorization helps us see how numbers can be broken down into their basic building blocks.

Square Root Calculation

Calculating the square root of a number means finding a value that, when multiplied by itself, gives the original number. For 225, we shall use the prime factors to achieve this.

• First, group the prime factors into pairs: (3 × 3) and (5 × 5).
• From each pair, take one number: 3 and 5.
• Multiply these numbers together: 3 × 5 = 15.

Thus, the square root of 225 through prime factorization is 15. This method is efficient because it simplifies the calculation by analyzing the structure of the number step by step. By pairing the factors, it ensures that you're finding a true square root, avoiding trial and error.

Verification of Results

Verification is a crucial step to ensure accuracy in any mathematical calculation. To confirm that 15 is indeed the square root of 225, it's important to multiply the result by itself to see if it returns to the original number.

• Simply compute: 15 × 15.
• The result will be 225, which matches the original number.

The successful verification solidifies our previous calculations and provides confidence in the solution. This step not only confirms accuracy but also reinforces understanding of how the initial factorization leads to the correct root. This crucial final check should never be overlooked to prevent simple errors and enhance learning.