Simplifying radicals often involves breaking down the radicand (the number inside the radical) into its prime factors. In the expression √14 + 8√3, the radical √14 cannot be simplified further as 14 has no perfect square factors other than 1. However, the radical √3 can be simplified. Let's explore how to fully simplify √14 + 8√3.
Understanding Simplification of Radicals
The fundamental principle in simplifying radicals lies in extracting perfect squares from the radicand. A perfect square is a number that results from squaring an integer. For instance, 9 is a perfect square because 3² = 9.
When dealing with radicals, the following property holds true:
√(a*b) = √a * √b
This property allows us to break down the radical into smaller parts, making it easier to identify and extract perfect squares.
Simplifying √14 + 8√3
-
Focus on √3: The radical √3 cannot be simplified further as 3 is a prime number and does not have any perfect square factors other than 1.
-
Simplifying 8√3: The coefficient 8 is outside the radical. It can be factored as 8 = 2 * 2 * 2 = 2² * 2. Therefore, 8√3 can be rewritten as 2²√3 * √2 = 2√3 * 2√2 = 4√6.
-
Combining terms: Now we have √14 + 8√3 = √14 + 4√6.
Final Result
The expression √14 + 8√3 can be fully simplified to √14 + 4√6. This simplification is possible due to the extraction of perfect squares from the radical √3 and the application of the radical property √(a*b) = √a * √b.
Note: The simplified expression √14 + 4√6 cannot be further simplified because √14 and √6 are both in their simplest form, with no common factors that can be extracted as perfect squares.
