_If x=−2, then √x⁶ = x³.

Verified step by step guidance

1. First, understand the expression \( \sqrt{x^6} \). This represents the square root of \( x^6 \).
2. Recall that the square root of a number is the same as raising that number to the power of \( \frac{1}{2} \). So, \( \sqrt{x^6} = (x^6)^{\frac{1}{2}} \).
3. Apply the power of a power property of exponents: \((a^m)^n = a^{m \cdot n}\). Therefore, \((x^6)^{\frac{1}{2}} = x^{6 \cdot \frac{1}{2}} = x^3\).
4. Now, substitute \( x = -2 \) into the expression \( x^3 \).
5. Calculate \( (-2)^3 \) to verify if it equals \( \sqrt{(-2)^6} \).

Key Concepts

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. In algebra, the square root function is denoted by the radical symbol (√). For example, √4 = 2 because 2 × 2 = 4. Understanding how square roots work is essential for solving equations involving them.

Exponents

Exponents represent repeated multiplication of a number by itself. For instance, x³ means x multiplied by itself three times. In the expression x⁶, the exponent indicates that x is multiplied by itself six times. Mastery of exponent rules, such as the power of a power and the product of powers, is crucial for simplifying expressions.

Negative Numbers and Even Powers

When dealing with negative numbers raised to even powers, the result is always positive. For example, (-2)² = 4 and (-2)⁶ = 64. This property is important when evaluating expressions involving negative bases, as it affects the outcome of square roots and other operations involving exponents.