Final answer:
To simplify the expression 2x(x - 3)⁻³ + x²(x - 3)⁻⅕, we find a common denominator for the exponents, rewrite the terms using this denominator, multiply through, and finally combine like terms to get the simplified form 5x²(x - 3)⁻⅕.
Step-by-step explanation:
The student asked how to simplify the algebraic expression 2x(x - 3)⁻³ + x²(x - 3)⁻⅕. This question is related to simplifying expressions with exponents, particularly negative and fractional exponents, and combining like terms.
First, we recognize that both terms have a common factor of (x - 3) raised to a negative fractional exponent. We must find a common denominator to combine them:
Common denominator: The least common exponent for (x - 3)⁻³ and (x - 3)⁻⅕ is ⁻⅕ because ⁻³ is equal to ⁻⅕ multiplied by 2/2.
Now, we can rewrite the expression using the common denominator:
2x(x - 3)⁻⅕ · 2/2 + x²(x - 3)⁻⅕
Multiply through by 2 for the first term:
4x²(x - 3)⁻⅕ + x²(x - 3)⁻⅕
Now, combine like terms:
(4x² + x²)(x - 3)⁻⅕
This simplifies further to:
5x²(x - 3)⁻⅕
This is the simplified form of the expression.