Find the square of each radical expression. See Example 2. √3x² + 4

Verified step by step guidance

1. Identify the expression to be squared: \(\sqrt{3x^2} + 4\).
2. Recall that squaring a sum follows the formula: \((a + b)^2 = a^2 + 2ab + b^2\).
3. Let \(a = \sqrt{3x^2}\) and \(b = 4\). Then, write the square as \((\sqrt{3x^2} + 4)^2 = (\sqrt{3x^2})^2 + 2 \times \sqrt{3x^2} \times 4 + 4^2\).
4. Simplify each term: \((\sqrt{3x^2})^2\) simplifies to \$3x^2\(, and \)4^2$ is \(16\).
5. Write the expanded form as \(3x^2 + 8 \sqrt{3x^2} + 16\).

Key Concepts

Radical Expressions

A radical expression involves roots, such as square roots, indicated by the radical symbol (√). Understanding how to interpret and manipulate these expressions is essential, especially recognizing that √(a) represents the principal square root of a.

Squaring a Radical Expression

Squaring a radical expression means raising it to the power of two. Since squaring and taking the square root are inverse operations, squaring √(expression) typically removes the radical, simplifying the expression to the value inside the root.

Algebraic Simplification

After squaring, simplifying the resulting algebraic expression involves combining like terms and applying exponent rules. For example, squaring √3x² + 4 requires careful distribution and simplification to correctly express the final polynomial.