Determine whether each statement is true or false. If false, correct the right side of the equation. (3x^2)^-1 = 3x^-2

Verified step by step guidance

1. Start by examining the left side of the equation: $(3x^2{)}^{-1}$.
2. Apply the property of exponents $(a^m{)}^n=a^{m\cdot n}$ to simplify: $(3x^2{)}^{-1}=3^{-1}\cdot (x^2{)}^{-1}$.
3. Simplify further: $3^{-1}=\frac{1}{3}$ and $(x^2{)}^{-1}=x^{-2}$, so $(3x^2{)}^{-1}=\frac{1}{3}\cdot x^{-2}$.
4. Now, examine the right side of the equation: $3x^{-2}$.
5. Compare both sides: $\frac{1}{3}\cdot x^{-2}$ is not equal to $3x^{-2}$, so the statement is false. The correct expression for $(3x^2{)}^{-1}$ is $\frac{1}{3}{x}^{-2}$.

Key Concepts

Exponents and Negative Exponents

Exponents represent repeated multiplication of a base number. A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent. For example, x^-n = 1/x^n. Understanding this concept is crucial for manipulating expressions involving negative exponents correctly.

Properties of Exponents

The properties of exponents include rules such as the product of powers, power of a power, and the power of a product. These rules help simplify expressions involving exponents. For instance, (a^m)^n = a^(m*n) and (ab)^n = a^n * b^n. Applying these properties correctly is essential for solving equations involving exponents.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves combining like terms and applying the rules of exponents to rewrite expressions in a more manageable form. This process often includes factoring, distributing, and reducing fractions. Mastery of simplification techniques is necessary to accurately assess the truth of algebraic statements and equations.