Question

the factor tree for 3,025 is shown.
what is the simplest form of \\(\sqrt{3,025}\\)?
\\(5^2 + (11^2)\\)
\\(5^2 (11^2)\\)
16
55

Explanation:

Step1: Analyze the factor tree

From the factor tree, we can see that \(3025 = 5\times5\times11\times11\). We can rewrite this as \(3025 = 5^{2}\times11^{2}\).

Step2: Simplify the square root

We know that \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (for \(a\geq0,b\geq0\)) and \(\sqrt{x^{2}} = x\) (for \(x\geq0\)). So, \(\sqrt{3025}=\sqrt{5^{2}\times11^{2}}\). Using the property of square roots, we can split this into \(\sqrt{5^{2}}\times\sqrt{11^{2}}\). Since \(\sqrt{5^{2}} = 5\) and \(\sqrt{11^{2}}=11\), then \(\sqrt{5^{2}}\times\sqrt{11^{2}}=5\times11 = 55\).

Now let's check the options:

• Option 1: \(5^{2}+(11^{2})=25 + 121=146

eq55\)

• Option 2: \(5^{2}(11^{2})=25\times121 = 3025\), this is the number itself, not the square root.
• Option 3: \(16

eq55\)

• Option 4: \(55\) which matches our calculation.

Answer:

The fourth option: 55