Question

Quotient rule with negative exponents: Prob Simplify.

(3x ³ + y ² + z ⁻⁶ /6x ⁻⁸ + y ⁻⁴
Write your answer using only positive exponents.

Answer 1

To simplify the expression with negative exponents, we convert the negative exponents to positive by taking the reciprocal and then apply the rules of dividing exponentials: dividing coefficients and subtracting exponents of like bases.

Step-by-step explanation:

The student has asked to simplify the expression (3x³ + y² + z⁻⁶) / (6x⁻⁸ + y⁻⁴) using only positive exponents. To achieve this, we apply the rule of division of exponentials, which states that we divide the coefficients and subtract the exponents of corresponding variables. However, in this question, subtraction of exponents is not required because the variables do not match between the numerator and the denominator.

Firstly, we recognize that a negative exponent indicates the reciprocal of that base raised to the positive exponent. Therefore, to convert x⁻⁸ to a positive exponent, we rewrite it as 1/x⁸. Similarly, y⁻⁴ becomes 1/y⁴ and z⁻⁶ becomes 1/z⁶.

To simplify the expression, we follow these steps:

1. Convert the negative exponents in the denominator to positive by taking the reciprocal: 6x⁻⁸ becomes 1/6x⁸ and y⁻⁴ becomes 1/y⁴.
2. Apply the reciprocal across the numerator: (3x³ + y² + 1/z⁶) × (x⁸/6) × y⁴.
3. Multiply the coefficient (3/6 reduces to 1/2) and apply the exponents to the variables: (1/2)x⁻⁵ + y¶ + xy⁴/z⁶

Note that in the last step, the terms are not combined because they do not have the same base or exponent; they remain separate terms within the simplified expression.