In Exercises 29–36, simplify and write the result in standard form. ____________ √1² − 4 ⋅ 0.5 ⋅ 5

Verified step by step guidance

1. Identify the expression under the square root, which is \(1^2 - 4 \cdot 0.5 \cdot 5\). This is a part of the quadratic formula's discriminant, but here we just need to simplify it.
2. Calculate each part inside the square root separately: first, square 1 to get \$1^2 = 1$.
3. Multiply the constants: \(4 \cdot 0.5 = 2\), then multiply by 5 to get \(2 \cdot 5 = 10\).
4. Substitute these values back into the expression under the square root: \$1 - 10$.
5. Simplify the expression under the square root to get \(-9\), then recognize that the square root of a negative number involves imaginary numbers, so write it as \(\sqrt{-9} = \sqrt{9} \cdot \sqrt{-1}\).

Key Concepts

Order of Operations

The order of operations dictates the sequence in which mathematical operations are performed to ensure consistent results. It follows the PEMDAS/BODMAS rules: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right). Applying this correctly is essential when simplifying expressions involving roots and arithmetic.

Square Root and Radicals

The square root of a number is a value that, when multiplied by itself, gives the original number. Simplifying square roots often involves evaluating the expression inside the radical first. Understanding how to handle radicals and simplify them is crucial for expressing results in standard form.

Standard Form of a Number

Standard form refers to expressing a number in a simplified, conventional way, often as a decimal or a simplified radical without complex components. In trigonometry and algebra, writing answers in standard form ensures clarity and uniformity, making it easier to interpret and use the results.