In Exercises 33–46, simplify each expression.____√36x⁴

Verified step by step guidance

1. Identify the expression under the square root: \( \sqrt{36x^4} \).
2. Recognize that the square root of a product can be separated into the product of square roots: \( \sqrt{36} \times \sqrt{x^4} \).
3. Calculate the square root of 36, which is a perfect square: \( \sqrt{36} = 6 \).
4. Simplify the square root of \( x^4 \) by recognizing it as \( (x^2)^2 \), so \( \sqrt{x^4} = x^2 \).
5. Combine the simplified parts to express the final simplified form: \( 6x^2 \).

Key Concepts

Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 36 is 6, since 6 × 6 = 36. Understanding square roots is essential for simplifying expressions involving radical signs.

Properties of Exponents

The properties of exponents govern how to manipulate expressions involving powers. For instance, the property that states x^(m/n) = n√(x^m) helps in simplifying expressions with variables raised to fractional powers. This is particularly useful when dealing with terms like x⁴ under a square root.

Simplifying Radicals

Simplifying radicals involves reducing the expression under the square root to its simplest form. This includes factoring out perfect squares from the radicand. For example, √(36x⁴) can be simplified by recognizing that both 36 and x⁴ are perfect squares, leading to a more straightforward expression.