Find the square of each radical expression. See Example 2. -√19

Verified step by step guidance

1. Identify the given expression: \(-\sqrt{19}\). This is a negative sign multiplied by the square root of 19.
2. Recall that squaring a number means multiplying the number by itself. So, we want to find \(\left(-\sqrt{19}\right)^2\).
3. Use the property of exponents: \(\left(a \cdot b\right)^2 = a^2 \cdot b^2\). Here, \(a = -1\) and \(b = \sqrt{19}\), so \(\left(-\sqrt{19}\right)^2 = (-1)^2 \cdot \left(\sqrt{19}\right)^2\).
4. Calculate each part separately: \((-1)^2 = 1\) because any real number squared is positive, and \(\left(\sqrt{19}\right)^2 = 19\) because squaring a square root cancels out the root.
5. Multiply the results: \(1 \cdot 19 = 19\). So, the square of \(-\sqrt{19}\) is 19.

Key Concepts

Square of a Number

Squaring a number means multiplying the number by itself. For any real number x, its square is x² = x × x. This operation is fundamental in algebra and helps simplify expressions involving radicals.

Properties of Square Roots

The square root of a number a, denoted √a, is a value that when squared gives a. Squaring a square root cancels the root, so (√a)² = a. This property is essential for simplifying expressions involving radicals.

Handling Negative Signs with Radicals

When squaring a negative radical like -√a, the negative sign is also squared. Since (-1)² = 1, the square of -√a is the same as the square of √a, resulting in a positive value. This ensures the final answer is positive.