Simplify.
$\sqrt{75}$
A. $3 \sqrt{5}$
B. $15 \sqrt{5}$
C. $25 \sqrt{3}$
D. $5 \sqrt{3}$
Answer (1)
Find the prime factorization of 75, which is 3 × 5 2 .
Rewrite the square root as 75 = 3 × 5 2 .
Simplify the square root: 3 × 5 2 = 5 3 .
The simplified expression is 5 3 .
Explanation
Understanding the problem We are asked to simplify the square root of 75, which means we need to find the largest perfect square that divides 75.
Prime factorization of 75 First, find the prime factorization of 75. We can write 75 as 3 × 25 , and since 25 is 5 2 , the prime factorization of 75 is 3 × 5 2 .
Rewriting the square root Now we can rewrite the square root of 75 as 75 = 3 × 5 2 .
Simplifying the square root Using the property of square roots, we can separate the factors: 3 × 5 2 = 3 × 5 2 . Since 5 2 = 5 , we have 75 = 5 3 .
Selecting the correct answer Comparing our simplified expression 5 3 with the given options, we see that it matches option D.
Examples
Square roots are used in many areas of math and science. For example, when calculating the distance between two points in a coordinate plane, we use the distance formula which involves square roots. If you wanted to find the distance between the points (1, 2) and (4, 6), you would calculate ( 4 − 1 ) 2 + ( 6 − 2 ) 2 = 3 2 + 4 2 = 9 + 16 = 25 = 5 . Simplifying square roots helps in expressing these distances in simplest form.
