Simplifying the Square Root of 216

In mathematics, simplification of expressions plays a crucial role in making complex equations more manageable. Today, we will explore how to simplify the square root of 216, a problem that involves the application of prime factorization and the properties of square roots. This process not only helps in evaluating the exact value but also in understanding the underlying principles of number theory.

Understanding the Problem

The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 4 is 2 because 2 * 2 4. However, for numbers like 216, finding the square root is not always straightforward, and simplification is often required to present the answer in a more comprehensible form.

Prime Factorization and Simplification

Prime factorization is a method of expressing a number as a product of its prime factors. For 216, the process is as follows:

Step 1: Identify Prime Factors
We start by dividing 216 by the smallest prime number, which is 2, and continue the process until we can only divide by 1.

Step 2: Perform the Division
216 ÷ 2 108
108 ÷ 2 54
54 ÷ 2 27
27 ÷ 3 9
9 ÷ 3 3
3 ÷ 3 1
So, 216 23 * 33.

Step 3: Simplify Using Square Root Properties
Knowing that sqrt{a * b} sqrt{a} * sqrt{b}, we can simplify further:

sqrt{216} sqrt{2^3 * 3^3} sqrt{2 * 2 * 2 * 3 * 3 * 3} sqrt{(2 * 2) * (3 * 3) * 2 * 3} sqrt{2^2 * 3^2 * 2 * 3}

Since sqrt{a^2} a, we can take out the squares:

sqrt{216} 2 * 3 * sqrt{2 * 3} 6 * sqrt{6}

This form, 6 * sqrt{6}, is considered simplified as it clearly shows the integer (6) multiplied by the square root of the non-square factor (6).

Conclusion

Summarizing the steps, we started with the prime factorization of 216, which is 23 * 33. We then used the property of square roots to factor out the squares, leading to the simplified form 6 * sqrt{6}. This simplified form not only makes the value easier to understand but also highlights the underlying mathematical principles.

It's important to note that while other forms can be derived by factoring further, they may not necessarily be simpler. The form 6 * sqrt{6} is considered the most simplified because it clearly separates the perfect squares from the non-square factors, making it more accessible for further mathematical operations.

In conclusion, simplifying the square root of 216 involves prime factorization and the application of square root properties, leading to the simplified form 6 * sqrt{6}.