Simplify The Square Root Of 216

To simplify the square root of 216, we need to break it down into its prime factors and look for perfect squares. The number 216 can be factored as 2 x 2 x 2 x 3 x 3 x 3. Put another way, 216 = 2³ x 3³.

When simplifying square roots, we look for pairs of the same prime factors because the square root of a squared number is the number itself. In this case, we have three 2s and three 3s. We can pair two of the 2s and two of the 3s, leaving us with one 2 and one 3 unpaired.

The square root of 216 can be written as √(2³ x 3³). We can simplify this by taking out the pairs from under the square root sign. The square root of 2² is 2, and the square root of 3² is 3. So, we have √(2² x 3²) x √(2 x 3).

This simplifies to 2 x 3 x √(2 x 3), which is 6 x √6. So, the simplified form of the square root of 216 is 6√6.

To further understand this process, let's look at the steps in detail:

1. Factor the number: Break down 216 into its prime factors: 2³ x 3³.
2. Identify pairs: Look for pairs of the same prime factors. In this case, we have two pairs: 2² and 3².
3. Simplify the square root: Take the square root of each pair and multiply them together. The square root of 2² is 2, and the square root of 3² is 3. So, we have 2 x 3 = 6.
4. Multiply by the remaining factors: Multiply the result from step 3 by the remaining factors under the square root sign. In this case, we have √(2 x 3) = √6.
5. Combine the results: Multiply the result from step 3 by the result from step 4. This gives us 6 x √6.

The final simplified form of the square root of 216 is 6√6.

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don't forget to note that simplifying square roots is a fundamental skill in algebra and is used in various mathematical applications. By breaking down numbers into their prime factors and looking for perfect squares, we can simplify complex expressions and make them easier to work with.

So, to summarize, the square root of 216 can be simplified to 6√6 by factoring the number into its prime factors, identifying pairs of the same prime factors, and simplifying the square root accordingly. This process is essential in algebra and is used in various mathematical applications.