QUESTION 4. Write the following as simply as possible without using negative indices and as a single fraction.

Let's re-examine Question 4, specifically part b), as there was a misread operator in the previous solution. QUESTION 4. Write the following as simply as possible without using negative indices and as a single fraction. a) (2x^-3)^3 y^-2x Step 1: Apply the power rule (ab)^n = a^n b^n and (a^m)^n = a^mn to the numerator. (2x^-3)^3 = 2^3 (x^-3)^3 = 8x^-9 Step 2: Substitute this back into the expression. 8x^-9 y^-2x Step 3: Simplify using the division rule a^m ÷ a^n = a^m-n. 8x^-9 y^-2x^1 = 8x^-9-1 y^-2 = 8x^-10 y^-2 Step 4: Rewrite the expression without negative indices and as a single fraction using a^-n = (1)/(a^n). 8x^-10 y^-2 = (8)/(x^10)y^2 The final answer is (8)/(x^10)y^2. b) x(2^-1y + y^-1x^-1) ÷ ((xy)^2)/(x^2) Step 1: Simplify the terms with negative exponents inside the parenthesis. 2^-1 = (1)/(2) y^-1 = (1)/(y) x^-1 = (1)/(x) The expression inside the parenthesis becomes (1)/(2)y + (1)/(y)(1)/(x) = (y)/(2) + (1)/(xy). Step 2: Distribute x into the parenthesis. x((y)/(2) + (1)/(xy)) = x · (y)/(2) + x · (1)/(xy) = (xy)/(2) + (1)/(y) Step 3: Simplify the divisor term. ((xy)^2)/(x^2) = (x^2 y^2)/(x^2) = y^2 Step 4: Perform the division. The expression is now ((xy)/(2) + (1)/(y)) ÷ y^2. To divide by y^2, multiply by its reciprocal, (1)/(y^2). ((xy)/(2) + (1)/(y)) × (1)/(y^2) Step 5: Distribute (1)/(y^2) into the parenthesis. (xy)/(2) × (1)/(y^2) + (1)/(y) × (1)/(y^2) = (x)/(2y) + (1)/(y^3) Step 6: Find a common denominator and combine the fractions. The least common multiple of 2y and y^3 is 2y^3. (x · y^2)/(2y · y^2) + (1 · 2)/(y^3 · 2) = (xy^2)/(2y^3) + (2)/(2y^3) Step 7: Add the fractions. (xy^2 + 2)/(2y^3) The final answer is (xy^2 + 2)/(2y^3). What's next?