Question

which expression is equivalent to ((3x^2)^{-1})?

1. (\frac{1}{3x^2})
2. (-3x^2)
3. (\frac{1}{9x^2})
4. (-9x^2)

Explanation:

Step1: Recall the negative exponent rule

The negative exponent rule states that \(a^{-n}=\frac{1}{a^{n}}\) for any non - zero real number \(a\) and integer \(n\). Also, \((ab)^{n}=a^{n}b^{n}\) (power of a product rule). For the expression \((3x^{2})^{-1}\), using the power of a product rule \((ab)^{-1}=a^{-1}b^{-1}\), we have \((3x^{2})^{-1}=3^{-1}\times(x^{2})^{-1}\).

Step2: Simplify each part

Using the negative exponent rule \(a^{-n}=\frac{1}{a^{n}}\), for \(3^{-1}\), we get \(3^{-1}=\frac{1}{3}\). For \((x^{2})^{-1}\), using the power of a power rule \((a^{m})^{n}=a^{mn}\), we have \((x^{2})^{-1}=x^{2\times(-1)} = x^{-2}=\frac{1}{x^{2}}\). Then, multiplying these two results together: \(3^{-1}\times(x^{2})^{-1}=\frac{1}{3}\times\frac{1}{x^{2}}=\frac{1}{3x^{2}}\).

Answer:

1. \(\frac{1}{3x^{2}}\)