Question

Rewrite the expression with positive exponents. $$\left(6 a^{-3}\right)^{3}$$

Short answer

The expression \( (6 a^{-3})^3 \) can be rewritten with positive exponents as \( \frac{216}{a^9} \).

Step-by-step solution

Identify Exponent Rules

Understanding exponent rules is key for solving this problem. A power of a power rule says that \( (a^m)^n = a^{m*n} \) and a negative exponent rule mentions that \( a^{-n} = \frac{1}{a^n} \). These rules will be used to simplify the expression.

Applying the power of a power rule

By applying the power of a power rule to the expression \( (6 a^{-3})^3 \), we get \(6^3 * a^{-3*3} = 216 * a^{-9}\).

Applying the negative exponent rule

By applying the negative exponent rule, \( a^{-9} = \frac{1}{a^9}\). Replace \( a^{-9} \) with \( \frac{1}{a^9} \) in the expression obtained from Step 2. We get: \( 216 * \frac{1}{a^9}\).