Find each root. $\(\sqrt{25k^4m^2}\)$

Verified step by step guidance

1. Identify the expression inside the square root: \(\sqrt{25k^{4}m^{2}}\).
2. Recall the property of square roots that \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\), so rewrite the expression as \(\sqrt{25} \cdot \sqrt{k^{4}} \cdot \sqrt{m^{2}}\).
3. Find the square root of each part separately: \(\sqrt{25}\), \(\sqrt{k^{4}}\), and \(\sqrt{m^{2}}\).
4. Use the rule \(\sqrt{x^{2}} = |x|\) to simplify the variable terms, so \(\sqrt{k^{4}} = |k^{2}|\) and \(\sqrt{m^{2}} = |m|\).
5. Combine all simplified parts to express the root as \(5 \cdot |k^{2}| \cdot |m|\).

Key Concepts

Square Root of a Product

The square root of a product can be expressed as the product of the square roots of each factor. For example, √(ab) = √a × √b. This property allows us to simplify complex expressions by breaking them into simpler parts.

Square Root of Variables with Exponents

When taking the square root of variables with exponents, divide the exponent by 2. For instance, √(x^n) = x^(n/2). This helps simplify expressions involving powers, especially when the exponent is even.

Simplifying Square Roots of Perfect Squares

A perfect square is a number or expression that is the square of an integer or variable expression. The square root of a perfect square is the base itself, such as √25 = 5 or √(k^4) = k^2, which simplifies the expression significantly.