Simplify The Square Root Of 180 In 3 Easy Steps

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Simplifying the square root of 180 can be a daunting task for many, but with a clear and step-by-step approach, it can be easily achieved. In this article, we will break down the process into 3 easy steps, making it accessible to everyone.

The square root of 180 is a value that, when multiplied by itself, gives the product 180. This value is often denoted by the symbol √180. Now, let's dive into the 3 easy steps to simplify it.

Step 1: Factorize 180 into Prime Numbers

To simplify the square root of 180, we first need to factorize 180 into its prime numbers. Prime numbers are numbers that are divisible only by 1 and themselves. The prime factorization of 180 is:

180 = 2 × 2 × 3 × 3 × 5

We can write this in a more concise way:

180 = 2^2 × 3^2 × 5

Now that we have the prime factorization of 180, we can proceed to the next step.

Understanding Prime Factorization

Before we move on, it's essential to understand why prime factorization is important in simplifying square roots. Prime factorization helps us identify the factors of a number and group them in a way that makes it easier to find the square root.

In the case of 180, we have two pairs of identical prime factors: 2^2 and 3^2. This is crucial in simplifying the square root, as we will see in the next step.

Step 2: Identify and Group the Perfect Squares

Now that we have the prime factorization of 180, we can identify and group the perfect squares. Perfect squares are numbers that can be expressed as the square of an integer. In this case, we have two perfect squares: 2^2 and 3^2.

We can rewrite the prime factorization of 180 as:

180 = (2^2 × 3^2) × 5

By grouping the perfect squares, we can simplify the square root of 180.

What are Perfect Squares?

Perfect squares are numbers that can be expressed as the square of an integer. For example, 4 is a perfect square because it can be expressed as 2^2. Similarly, 9 is a perfect square because it can be expressed as 3^2.

In the case of 180, we have two perfect squares: 2^2 and 3^2. By identifying and grouping these perfect squares, we can simplify the square root of 180.

Step 3: Simplify the Square Root

Now that we have identified and grouped the perfect squares, we can simplify the square root of 180. We can rewrite the square root of 180 as:

√180 = √(2^2 × 3^2 × 5)

Using the property of square roots that states √(a × b) = √a × √b, we can simplify the expression:

√180 = √(2^2) × √(3^2) × √5

= 2 × 3 × √5

= 6√5

Therefore, the simplified square root of 180 is 6√5.

Final Answer

The final answer is 6√5. This is the simplified form of the square root of 180.

In conclusion, simplifying the square root of 180 can be achieved in 3 easy steps: factorizing 180 into prime numbers, identifying and grouping the perfect squares, and simplifying the square root. By following these steps, you can simplify the square root of 180 and many other numbers.

We hope this article has helped you understand the concept of simplifying square roots. If you have any questions or need further clarification, please don't hesitate to ask.