Determine whether each statement is true or false. If false, correct the right side of the equation. (3x²)^-1 = 3x^-2

Verified step by step guidance

1. Start by examining the left side of the equation: \((3x^2)^{-1}\). Recall that a negative exponent means taking the reciprocal, so \((a)^{-1} = \frac{1}{a}\).
2. Rewrite the left side using this rule: \((3x^2)^{-1} = \frac{1}{3x^2}\).
3. Now, look at the right side of the equation: \$3x^{-2}\(. Using the negative exponent rule, \)x^{-2} = \frac{1}{x^2}\(, so \)3x^{-2} = 3 \times \frac{1}{x^2} = \frac{3}{x^2}$.
4. Compare the two expressions: the left side is \(\frac{1}{3x^2}\) and the right side is \(\frac{3}{x^2}\). Since these are not equal, the original statement is false.
5. To correct the right side so the equation holds true, write it as \(\frac{1}{3x^2}\) or equivalently \(\frac{1}{3} x^{-2}\).

Key Concepts

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the positive exponent. For example, x^(-n) equals 1 divided by x^n. Understanding this helps in rewriting expressions with negative powers correctly.

Exponent Rules for Powers of Products

When raising a product to an exponent, apply the exponent to each factor separately: (ab)^n = a^n * b^n. This rule is essential for correctly expanding expressions like (3x^2)^-1.

Simplifying Algebraic Expressions

Simplification involves applying exponent rules and combining like terms to rewrite expressions in a clearer or more standard form. This skill is necessary to verify or correct equations involving exponents.