Question

write the quotient in its simplest form.\\(
\frac{\sqrt{338}}{\sqrt{72}}
\\)
\\(
\frac{\sqrt{338}}{\sqrt{72}} = \square
\\)
(type an exact answer, using radicals as needed.)

Explanation:

Step1: Use the quotient rule for radicals

The quotient rule for radicals states that $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ (where $a\geq0$ and $b>0$). So we can rewrite $\frac{\sqrt{338}}{\sqrt{72}}$ as $\sqrt{\frac{338}{72}}$.

Step2: Simplify the fraction inside the radical

Simplify $\frac{338}{72}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 2. So $\frac{338\div2}{72\div2}=\frac{169}{36}$. Now our expression becomes $\sqrt{\frac{169}{36}}$.

Step3: Take the square root of the numerator and denominator

We know that $\sqrt{\frac{169}{36}}=\frac{\sqrt{169}}{\sqrt{36}}$ (by the quotient rule for radicals again). And $\sqrt{169} = 13$ and $\sqrt{36}=6$. So putting it together, we get $\frac{13}{6}$. Wait, no, wait. Wait, did I make a mistake? Wait, 338 divided by 72: 338 is 2×13×13, 72 is 2×36 = 2×6×6. Wait, maybe I messed up the simplification. Let's redo step 2. Let's factor 338 and 72. 338 = 2×13², 72 = 2×6². So $\frac{338}{72}=\frac{2\times13^{2}}{2\times6^{2}}$. Then the 2s cancel out, so we have $\frac{13^{2}}{6^{2}}$. Then $\sqrt{\frac{13^{2}}{6^{2}}}=\frac{13}{6}$? Wait, but that seems too easy. Wait, no, wait 338 is 13²×2? 13² is 169, 169×2 is 338, yes. 72 is 6²×2, 6² is 36, 36×2 is 72. So when we have $\frac{\sqrt{338}}{\sqrt{72}}=\sqrt{\frac{338}{72}}=\sqrt{\frac{2\times13^{2}}{2\times6^{2}}}=\sqrt{\frac{13^{2}}{6^{2}}}=\frac{13}{6}$. Wait, but that's a rational number. Is that correct? Let's check: 13/6 squared is 169/36, and 338/72 is (338÷2)/(72÷2)=169/36. Yes, so that's correct. Wait, but maybe I misread the problem. Wait the problem is $\frac{\sqrt{338}}{\sqrt{72}}$. So yes, that simplifies to 13/6. Wait, but let me confirm the factoring again. 13×13 is 169, 169×2 is 338. 6×6 is 36, 36×2 is 72. So when we take the square root of (338/72), it's the square root of (169/36), which is 13/6. So that's the simplest form.

Answer:

$\frac{13}{6}$