Use the product rule to simplify the radical. $$ \sqrt{117} $$

Question

UpStudy AI · Accepted Answer

To simplify the radical $$\sqrt{117}$$ using the product rule, follow these steps:

**Step 1: Factor the Number Inside the Radical**

Find the prime factors of 117 to identify any perfect squares.

$$
117 = 9 \times 13
$$

Here, 9 is a perfect square because $$9 = 3^2$$.

**Step 2: Apply the Product Rule for Radicals**

The product rule states that $$\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$$. Apply this to $$\sqrt{117}$$:

$$
\sqrt{117} = \sqrt{9 \times 13} = \sqrt{9} \times \sqrt{13}
$$

**Step 3: Simplify the Square Root of the Perfect Square**

$$
\sqrt{9} = 3
$$

**Step 4: Combine the Results**

$$
\sqrt{117} = 3 \times \sqrt{13} = 3\sqrt{13}
$$

**Final Answer:**
$$
\sqrt{117} = 3\sqrt{13}
$$
$$ \sqrt{117} = 3\sqrt{13} $$

Use the product rule to simplify the radical.