In Exercises 33–46, simplify each expression._____−√100x⁶

Verified step by step guidance

1. Identify the expression under the square root: \( \sqrt{100x^6} \).
2. Recognize that \( 100x^6 \) can be rewritten as \( (10x^3)^2 \) because \( 10^2 = 100 \) and \( (x^3)^2 = x^6 \).
3. Apply the property of square roots: \( \sqrt{a^2} = |a| \), which means \( \sqrt{(10x^3)^2} = |10x^3| \).
4. Since the original expression is \( -\sqrt{100x^6} \), apply the negative sign to the result: \( -|10x^3| \).
5. Consider the absolute value: \( |10x^3| = 10|x^3| \), so the expression simplifies to \( -10|x^3| \).

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 100 is 10, since 10 × 10 = 100. In algebra, understanding how to simplify square roots, especially with variables, is crucial for solving expressions effectively.

Properties of Exponents

Exponents represent repeated multiplication of a base number. The property of exponents states that x^a * x^b = x^(a+b) and x^a / x^b = x^(a-b). When simplifying expressions involving variables raised to powers, it is essential to apply these properties correctly to combine or reduce terms.

Simplifying Radical Expressions

Simplifying radical expressions involves reducing the expression to its simplest form, which often includes factoring out perfect squares from under the radical. For instance, √(x^6) simplifies to x^3, as x^3 * x^3 = x^6. This process is vital for making expressions easier to work with in algebra.