Simplify $\sqrt{20}$. The simplified expression is $\square$.
Answer (1)
Find the prime factorization of 20: 20 = 2 2 ⋅ 5 .
Rewrite the square root: 20 = 2 2 ⋅ 5 .
Separate the factors: 2 2 ⋅ 5 = 2 2 ⋅ 5 .
Simplify: 20 = 2 5 . The simplified expression is 2 5 .
Explanation
Understanding the Problem We are asked to simplify the square root of 20, which means we want to find the simplest radical form of 20 .
Prime Factorization To simplify 20 , we first find the prime factorization of 20. We can express 20 as a product of its prime factors: 20 = 2 ⋅ 2 ⋅ 5 = 2 2 ⋅ 5 .
Rewriting the Square Root Now we rewrite the square root using the prime factorization: 20 = 2 2 ⋅ 5 .
Separating Factors Using the property a ⋅ b = a ⋅ b , we can separate the factors: 2 2 ⋅ 5 = 2 2 ⋅ 5 .
Simplifying the Perfect Square We simplify the perfect square: 2 2 = 2 .
Simplified Expression Finally, we write the simplified expression: 2 5 .
Final Answer Therefore, the simplified form of 20 is 2 5 .
Examples
Simplifying radicals is useful in many areas of mathematics, such as geometry and algebra. For example, when finding the distance between two points, you might end up with a radical that needs to be simplified. Suppose you are building a garden and need to calculate the length of a diagonal support beam. If the sides of the garden are 2 meters and 5 meters, the diagonal length is 2 2 + 5 2 = 29 meters. Simplifying radicals helps in expressing such lengths in their simplest form, making measurements and calculations easier.
