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

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Start by understanding the expression \((3x^{2})^{-1}\). The negative exponent means you take the reciprocal of the base, so \((a)^{-1} = \frac{1}{a}\).

Rewrite \((3x^{2})^{-1}\) as \(\frac{1}{3x^{2}}\). This means the entire quantity \(3x^{2}\) is in the denominator.

Now, consider the right side of the equation, which is \(3x^{-2}\). This can be rewritten as \(3 \times x^{-2} = 3 \times \frac{1}{x^{2}} = \frac{3}{x^{2}}\).

Compare the two expressions: \(\frac{1}{3x^{2}}\) on the left and \(\frac{3}{x^{2}}\) on the right. They are not equal because the numerator and denominator differ.

To correct the right side to be equivalent to the left, write it as \(\frac{1}{3x^{2}}\) or equivalently \(3^{-1} x^{-2}\). So the correct expression is \((3x^{2})^{-1} = 3^{-1} x^{-2}\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Negative Exponent Rule

The negative exponent rule states that a term with a negative exponent can be rewritten as the reciprocal with a positive exponent. For example, a^{-n} = 1/a^n. This rule helps simplify expressions involving negative powers.

Power of a Product Rule

When raising a product to a power, each factor inside the parentheses is raised to that power separately. Formally, (ab)^n = a^n * b^n. This rule is essential for correctly distributing exponents over multiplication.

Simplifying Exponential Expressions

Simplifying exponential expressions involves applying exponent rules systematically to rewrite expressions in simpler or equivalent forms. This includes combining like bases, applying power rules, and ensuring correct handling of negative exponents.

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