Questions: Simplify the expression using the properties of exponents. Expand any numerical portion of your answer and only include positive exponents. (3 x^(-3))^(-2)

Solution Steps

To simplify the expression \(\left(3 x^{-3}\right)^{-2}\), we will use the properties of exponents. Specifically, we will apply the power of a power property \((a^m)^n = a^{m \cdot n}\) and the property of negative exponents \(a^{-m} = \frac{1}{a^m}\).

1. Apply the power of a power property to both the numerical and variable parts.
2. Simplify the resulting expression to ensure all exponents are positive.

The given expression is: \[ \left(3 x^{-3}\right)^{-2} \]

Using the power of a power property \((a^m)^n = a^{mn}\), we can distribute the exponent \(-2\) to both the base \(3\) and the exponent of \(x\): \[ \left(3 x^{-3}\right)^{-2} = 3^{-2} \cdot (x^{-3})^{-2} \]

First, simplify \(3^{-2}\): \[ 3^{-2} = \frac{1}{3^2} = \frac{1}{9} \]

Next, simplify \((x^{-3})^{-2}\) using the power of a power property: \[ (x^{-3})^{-2} = x^{-3 \cdot (-2)} = x^6 \]

Now, combine the simplified terms: \[ 3^{-2} \cdot x^6 = \frac{1}{9} \cdot x^6 = \frac{x^6}{9} \]

Final Answer