Question

simplify.
\sqrt{\frac{16}{64}}
be sure to write your answer in simplest form.

Explanation:

Step1: Simplify the fraction inside the square root

First, simplify the fraction \(\frac{16}{64}\). We can divide both the numerator and the denominator by 16. So, \(\frac{16\div16}{64\div16}=\frac{1}{4}\)? Wait, no, wait. Wait, 16 divided by 16 is 1, 64 divided by 16 is 4? Wait, no, 16 times 4 is 64? Wait, 16 times 4 is 64? Wait, 164=64? Yes. Wait, but also, 16/64 can be simplified by dividing numerator and denominator by 16: 16÷16=1, 64÷16=4? Wait, no, 64÷16 is 4? Wait, 164=64, yes. Wait, but also, 16/64 = 1/4? Wait, no, 16/64 = 1/4? Wait, 16 divided by 16 is 1, 64 divided by 16 is 4? Wait, no, 64 divided by 16 is 4? Wait, 16*4=64, so 64÷16=4. So 16/64=1/4? Wait, no, 16/64 = (16÷16)/(64÷16)=1/4? Wait, no, 16/64 = 1/4? Wait, 16 divided by 64 is 0.25, and 1/4 is 0.25, yes. Wait, but also, 16/64 can be simplified as 1/4, but also, 16/64 = 1/4, or wait, 16 and 64 are both perfect squares. Wait, maybe another approach: use the property of square roots \(\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\) for \(a\geq0\) and \(b>0\). So, \(\sqrt{\frac{16}{64}}=\frac{\sqrt{16}}{\sqrt{64}}\). Then, \(\sqrt{16}=4\) because \(4^2=16\), and \(\sqrt{64}=8\) because \(8^2=64\). So then, \(\frac{4}{8}\). Then, simplify \(\frac{4}{8}\) by dividing numerator and denominator by 4: \(\frac{4\div4}{8\div4}=\frac{1}{2}\). Wait, that's a better approach. Let's do that.

Step1: Apply the square root of a fraction property

Using the property \(\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\) (where \(a = 16\), \(b = 64\), \(a\geq0\), \(b>0\)), we get:
\(\sqrt{\frac{16}{64}}=\frac{\sqrt{16}}{\sqrt{64}}\)

Step2: Simplify the square roots

Simplify \(\sqrt{16}\) and \(\sqrt{64}\). We know that \(4^2 = 16\), so \(\sqrt{16}=4\). Also, \(8^2 = 64\), so \(\sqrt{64}=8\). So now we have:
\(\frac{\sqrt{16}}{\sqrt{64}}=\frac{4}{8}\)

Step3: Simplify the fraction

Simplify \(\frac{4}{8}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 4. So:
\(\frac{4\div4}{8\div4}=\frac{1}{2}\)

Answer:

\(\frac{1}{2}\)