Solving the Square Root of 27: A Step-by-Step Guide

Solving the square root of 27 can be approached in various ways, all of which yield the same result. This guide walks you through the process of simplifying and finding the square root of 27, using different mathematical techniques such as prime factorization and basic algebraic principles.

Method 1: Prime Factorization

Solving the square root of 27 using prime factorization involves breaking down the radicand (the number under the square root) into its prime factors and then simplifying the expression.

First, factor 27 into its prime components:

27 3 × 3 × 3 33

Next, use the properties of square roots to simplify the expression:

sqrt{27} sqrt{3^3} sqrt{3^2} times sqrt{3} 3sqrt{3}

The final answer is:

3sqrt{3} or approximately 5.196.

Method 2: Simplifying Using Algebraic Properties

The square root of 27 can also be simplified using the algebraic property that the square root of a product is the product of the square roots: sqrt{xy} sqrt{x}sqrt{y}.

Express 27 as a product of its factors:

27 9 × 3

Apply the square root property:

sqrt{27} sqrt{9 × 3} sqrt{9} × sqrt{3}

Knowing that the square root of 9 is 3, we get:

sqrt{9} × sqrt{3} 3 × sqrt{3}

The final answer is again:

3sqrt{3}, which is approximately 5.196.

Additional Approach: Using Decimals

Using the decimal approximation, the square root of 3 is approximately 1.73. Therefore:

Substitute the value of sqrt{3} in the simplified form:

sqrt{27} 3 × sqrt{3} 3 × 1.73

Compute the multiplication:

3 × 1.73 5.19

Thus, the square root of 27 is approximately 5.19.

Conclusion

When solving the square root of 27, you can use prime factorization, algebraic properties, or decimal approximations. Each method leads to the same result, providing a clear and accurate way to find the square root. Understanding and applying these techniques enhances your problem-solving skills in mathematics.

Key Takeaways:

The square root of 27 is 3sqrt{3} or approximately 5.196. Prime factorization of 27 is 3^3. The square root of 9 is 3.