Simplify: [tex]\sqrt{\frac{25}{441}}[/tex]
The prime factorization of 25 is $\square$ .
The prime factorization of 441 is $\square$
The expression [tex]\sqrt{\frac{25}{441}}[/tex] in simplest form is $\square$
Answer (1)
Find the prime factorization of 25, which is 5 2 .
Find the prime factorization of 441, which is 3 2 × 7 2 .
Take the square root of the numerator and the denominator separately: 441 25 = 441 25 = 21 5 .
The simplified expression is 21 5 .
Explanation
Problem Analysis We are asked to simplify the square root of a fraction, 441 25 . To do this, we will first find the prime factorization of the numerator and the denominator. Then, we will take the square root of both the numerator and the denominator separately.
Prime Factorization of 25 The prime factorization of 25 is 5 × 5 = 5 2 .
Prime Factorization of 441 The prime factorization of 441 is 3 × 3 × 7 × 7 = 3 2 × 7 2 .
Rewriting the Expression Now we can rewrite the original expression using the prime factorizations: 441 25 = 3 2 × 7 2 5 2 .
Separating the Square Roots Next, we take the square root of the numerator and the denominator separately: 3 2 × 7 2 5 2 = 3 2 × 7 2 5 2 .
Calculating the Square Roots The square root of 5 2 is 5, and the square root of 3 2 × 7 2 is 3 × 7 = 21 . Therefore, we have 3 2 × 7 2 5 2 = 21 5 .
Final Answer Thus, the simplified form of the expression is 21 5 .
Examples
Imagine you are tiling a square area and want to use square tiles. If the area is 441 25 square meters, then the side length of the square area is 441 25 meters. Simplifying this gives you 21 5 meters, which tells you the length of each side of the square area. This is also the side length of the largest square tile you can use to perfectly cover the area without cutting any tiles, assuming the side lengths of the tiles must be a rational number.
