The factor tree for 1,764 is shown.
What is the simplest form of $\sqrt{1,764}$ ?
A. 21
B. 42
C. $3^2(7^2)$
D. $2^2(3^2)(7^2)$
Answer (2)
The simplest form of 1 , 764 is 42 , which is derived from its prime factorization of 1764 = 2 2 × 3 2 × 7 2 . By simplifying the square root, we can calculate that 2 × 3 × 7 = 42 . Therefore, the answer is option B: 42.
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Find the prime factorization of 1,764: 1764 = 2 2 × 3 2 × 7 2 .
Take the square root of the prime factorization: 1764 = 2 2 × 3 2 × 7 2 .
Simplify the square root: 2 2 × 3 2 × 7 2 = 2 × 3 × 7 .
Calculate the product: 2 × 3 × 7 = 42 .
Explanation
Problem Analysis We are asked to find the simplest form of 1 , 764 using the factor tree (not shown).
Prime Factorization The prime factorization of 1,764 can be found from the factor tree (not shown) and is 1764 = 2 2 × 3 2 × 7 2 .
Square Root of Prime Factors To find the simplest form of 1 , 764 , we take the square root of the prime factorization: 1 , 764 = 2 2 × 3 2 × 7 2 .
Simplifying the Square Root Simplifying the square root, we get 2 2 × 3 2 × 7 2 = 2 × 3 × 7 .
Calculating the Product Calculating the product, we have 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 feet, you would need to find the square root of the area. In this case, the side length of the garden would be 1 , 764 = 42 feet. Then, using the Pythagorean theorem, you can find the diagonal length, which involves further calculations with square roots.
