Final answer:
The expression (a⁵ * a / a⁻³)⁻¹ simplifies to 1/a³. This is done by using the rules for exponents where you add exponents when multiplying with the same base and subtract them when dividing, also recalling negative exponents denote reciprocal values.
Step-by-step explanation:
To represent the expression (a⁵ * a / a⁻³)⁻¹, we will follow the rules associated with exponents. Remember, negative exponents essentially move the base from the numerator to the denominator, or vice versa, converting multiplication into division and division into multiplication. Thus, a negative exponent indicates taking the reciprocal of the base raised to the positive equivalent of that exponent.
First, we combine the exponents in the numerator by adding them, since they have the same base, according to the rule stated in Eq. A.8. So we have a⁵ * a = a⁵¹ = a⁶. Now we will deal with the denominator a⁻³, remembering that negative exponents are equivalent to the reciprocal of the base raised to the positive exponent (a⁻³ = 1/a³). We now have the expression a⁶/a³.
To divide the exponents, we subtract the exponent in the denominator from the exponent in the numerator: 6 - 3 = 3. So a⁶/a³ = a³. Lastly, we deal with the final negative exponent (a³)⁻¹. Applying the rule for negative exponents again, this can be simplified to 1/a³.
In conclusion, the expression (a⁵ * a / a⁻³)⁻¹ simplifies to 1/a³.