Simplifying radicals in algebra: example of √125n = 5√5n.

Problem:

Simplify the radical expression:
$$
\sqrt{125n}
$$

Solution:

The goal is to simplify the square root expression by factoring out perfect squares from the radicand (the expression inside the square root). Let's break it down step by step.

#### Step 1: Factorize the number under the square root
The number inside the square root is \(125\). We need to factorize \(125\) into its prime factors:
$$
125 = 5 \cdot 25
$$
Next, we recognize that \(25\) is a perfect square:
$$
25 = 5^2
$$
So, we can rewrite \(125\) as:
$$
125 = 5 \cdot 5^2
$$

#### Step 2: Rewrite the expression
Now, substitute the factorization back into the original expression:
$$
\sqrt{125n} = \sqrt{5 \cdot 5^2 \cdot n}
$$

#### Step 3: Use the property of square roots
The square root of a product can be written as the product of the square roots:
$$
\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}
$$
Applying this property:
$$
\sqrt{5 \cdot 5^2 \cdot n} = \sqrt{5^2} \cdot \sqrt{5} \cdot \sqrt{n}
$$

#### Step 4: Simplify the perfect square
We know that:
$$
\sqrt{5^2} = 5
$$
So:
$$
\sqrt{5^2} \cdot \sqrt{5} \cdot \sqrt{n} = 5 \cdot \sqrt{5} \cdot \sqrt{n}
$$

#### Step 5: Combine the terms
Combine the constants and variables under the square root:
$$
5 \cdot \sqrt{5} \cdot \sqrt{n} = 5\sqrt{5n}
$$

Final Answer:

The simplified form of the expression is:
$$
\boxed{5\sqrt{5n}}
$$

Explanation:

- Step 1: Factorize the number \(125\) into its prime factors and identify perfect squares.
- Step 2: Rewrite the expression using the factorization.
- Step 3: Apply the property of square roots to separate the factors.
- Step 4: Simplify the perfect square.
- Step 5: Combine the results to get the final simplified form.

This method ensures that the expression is simplified correctly by extracting all possible perfect squares from the radicand.