Simplify the following radical expression.
$\sqrt{20}$
A. $2 \sqrt{5}$
B. $5 \sqrt{2}$
C. $4 \sqrt{5}$
D. $10 \sqrt{5}$
Answer (1)
20 can be simplified by expressing 20 as a product of its prime factors, 20 = 2 2 × 5 . Then, 20 = 2 2 × 5 = 2 5 . The simplified expression is 2 5 . Therefore, the correct answer is 2 5 .
Explanation
Understanding the problem We are asked to simplify the radical expression 20 . This means we want to find the largest perfect square that divides 20 and then take its square root.
Prime factorization We can write 20 as a product of its prime factors: 20 = 2 × 2 × 5 = 2 2 × 5 .
Rewriting the square root Now we can rewrite the square root of 20 using the prime factorization: 20 = 2 2 × 5 .
Simplifying the square root We can simplify the square root by taking out the perfect square factor, 2 2 , from under the radical: 2 2 × 5 = 2 2 × 5 = 2 × 5 = 2 5 .
Selecting the correct answer Comparing our simplified expression, 2 5 , with the given options, we see that it matches option A.
Examples
Radical expressions are useful in many areas, such as calculating distances using the Pythagorean theorem. For example, if you have a right triangle with legs of length 2 and 4, the length of the hypotenuse is 2 2 + 4 2 = 4 + 16 = 20 = 2 5 . Simplifying radical expressions allows us to express these lengths in a more manageable form.
