Question

the factor tree for 1,764 is shown. what is the simplest form of \\(\sqrt{1,764}\\)? \\(\circ\\) 21 \\(\circ\\) 42 \\(\circ\\) \\(3^2(7^2)\\) \\(\circ\\) \\(2^2(3^2)(7^2)\\)

Explanation:

Step1: Prime Factorization from Tree

From the factor tree, \(1764 = 2\times2\times3\times3\times7\times7\). Rewrite as exponents: \(1764 = 2^2\times3^2\times7^2\).

Step2: Simplify Square Root

Use \(\sqrt{ab}=\sqrt{a}\sqrt{b}\) and \(\sqrt{x^2}=x\). So \(\sqrt{1764}=\sqrt{2^2\times3^2\times7^2}=\sqrt{2^2}\times\sqrt{3^2}\times\sqrt{7^2}=2\times3\times7 = 42\). Also, check options: \(2^2(3^2)(7^2)\) is the prime factorization, but the simplest form (numerical) is 42. Wait, no—wait, the question says "simplest form of \(\sqrt{1764}\)". Wait, let's re - evaluate. Wait, the options: 21, 42, \(3^2(7^2)\), \(2^2(3^2)(7^2)\). Wait, \(\sqrt{1764}\): let's compute \(42\times42 = 1764\), so \(\sqrt{1764}=42\). Let's check the prime factors: \(1764 = 2^2\times3^2\times7^2\), so \(\sqrt{1764}=\sqrt{2^2\times3^2\times7^2}=2\times3\times7 = 42\). So the correct option is 42.

Answer:

42 (the option is "42" among the given choices, so the answer is the option with "42", i.e., the second option: 42)