Question

simplify.
sqrt{\frac{125}{45}}
be sure to write your answer in simplest form.

Explanation:

Step1: Simplify the fraction inside the square root.

We have \(\frac{125}{45}\). First, find the greatest common divisor (GCD) of 125 and 45. The factors of 125 are \(1, 5, 25, 125\) and the factors of 45 are \(1, 3, 5, 9, 15, 45\). The GCD is 5. Divide both the numerator and the denominator by 5: \(\frac{125\div5}{45\div5}=\frac{25}{9}\). So now we need to simplify \(\sqrt{\frac{25}{9}}\).

Step2: Use the property of square roots \(\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\) (where \(a\geq0\) and \(b > 0\)).

Applying this property, \(\sqrt{\frac{25}{9}}=\frac{\sqrt{25}}{\sqrt{9}}\). We know that \(\sqrt{25} = 5\) because \(5\times5 = 25\) and \(\sqrt{9}=3\) because \(3\times3 = 9\). So \(\frac{\sqrt{25}}{\sqrt{9}}=\frac{5}{3}\).

Answer:

\(\frac{5}{3}\)