Questions: Simplify the expression using the properties of exponents. Answer using only nonnegative exponents. (3 cm)^-4

Solution Steps

To simplify the expression \((3 \mathrm{~cm})^{-4}\) using the properties of exponents, we need to apply the rule that \(a^{-n} = \frac{1}{a^n}\). This will convert the negative exponent into a positive one.

1. Recognize that the negative exponent indicates a reciprocal.
2. Apply the rule \(a^{-n} = \frac{1}{a^n}\) to the given expression.

We start with the expression \((3 \mathrm{~cm})^{-4}\). The negative exponent indicates that we need to take the reciprocal of the base raised to the positive exponent.

Using the property of exponents, we rewrite the expression as: \[ (3 \mathrm{~cm})^{-4} = \frac{1}{(3 \mathrm{~cm})^{4}} \]

Next, we calculate \((3 \mathrm{~cm})^{4}\): \[ (3 \mathrm{~cm})^{4} = 3^{4} \cdot \mathrm{cm}^{4} = 81 \mathrm{~cm}^{4} \]

Now, substituting back into our expression, we have: \[ (3 \mathrm{~cm})^{-4} = \frac{1}{81 \mathrm{~cm}^{4}} \approx 0.0123 \]

Final Answer