Final answer:
The question involves correcting an algebraic function using positive and negative exponents, as well as fractional exponents. Positive exponents imply multiplication, negative exponents denote division, and fractional exponents represent roots. The function already uses fractional exponents and does not require further correction without additional instructions.
Step-by-step explanation:
The question asks us to correct the function h(x) = ((2x² - x + 6)²/3) / (2x² - 2x + 9) using positive and negative exponents and fractional exponents. When manipulating expressions with exponents, remember that positive exponents indicate multiplication of the base number by itself a number of times equal to the exponent, while negative exponents imply division by the base raised to the corresponding positive exponent. Fractional exponents, such as the square root being represented as an exponent of 1/2, denote roots; for example, x² is the same as √x.
Here's how you would apply these rules to the given function:
- Understand that the exponent 2/3 on the numerator suggests the cube root of the quantity squared. Thus, the numerator can be rewritten as ∙(2x² - x + 6)².
- Since the denominator does not have an exponent, we treat it as if it has an exponent of 1, which adheres to the identity rule that any non-zero number to the power of 1 is itself.
- If we needed to simplify further, we would utilize properties of exponents to combine like terms, keeping in mind that subtracting exponents is used for division, and flipping a term with a negative exponent to the denominator.
Please note that in this specific case, there is no further simplification possible without more context or specific instructions on how to transform the function.