What is the simplest form of [tex]$\sqrt{1,764}$[/tex]?
A. 21
B. 42
C. [tex]$3^2\left(7^2\right)$[/tex]
D. [tex]$2^2\left(3^2\right)\left(7^2\right)$[/tex]
Answer (2)
Find the prime factorization of 1,764: 1764 = 2 2 × 3 2 × 7 2 .
Take the square root: 1 , 764 = 2 2 × 3 2 × 7 2 .
Simplify the square root: 2 2 × 3 2 × 7 2 = 2 × 3 × 7 .
Calculate the final value: 2 × 3 × 7 = 42 .
Explanation
Problem Analysis We are asked to find the simplest form of 1 , 764 given the factor tree for 1,764. The factor tree will help us find the prime factorization of 1,764, which we can then use to simplify the square root.
Prime Factorization From the factor tree (not shown, but implied), we can find the prime factorization of 1,764. The prime factorization is 1764 = 2 2 × 3 2 × 7 2 .
Square Root Now, we take the square root of 1,764: 1 , 764 = 2 2 × 3 2 × 7 2 .
Simplifying the Square Root We simplify the square root by taking the square root of each factor: 2 2 × 3 2 × 7 2 = 2 2 × 3 2 × 7 2 = 2 × 3 × 7 .
Calculating the Final Value Finally, we multiply the factors together: 2 × 3 × 7 = 6 × 7 = 42 .
Final Answer Therefore, the simplest form of 1 , 764 is 42.
Examples
Understanding square roots is essential in various fields, such as engineering and physics. For instance, when calculating the length of the diagonal of a square garden with an area of 1,764 square meters, you would need to find the square root of 1,764 to determine the side length of the garden. Then, using the Pythagorean theorem, you can find the diagonal length. This concept is also used in financial analysis to calculate investment growth rates and in computer graphics to determine distances and sizes.
The simplest form of 1764 is 42 . Therefore, the correct answer is option B.
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