Expand (1+0.5x)^4, simplifying the coefficients.

Step 1. Draw Pascal's triangle to find the coefficients. 1; 1 2 1 ; 1 3 3 1 ; 1 4 6 4 1. As you can see, each row starts and finishes with 1. The numbers in between are worked out by adding the two numbers on the row above. For this question, we will use the 4th row 1 4 6 4 1 since the expression is raised to the power of 4. This expansion will have 5 expressions.Step 2. For each term, both 1 and 0.5 are raised to the powers 0 to 4, where the sum of the powers adds up to 4. The power of x is increased from 0 to 4 as the term progresses. (1 + 0.5) ⁴ = 1(1)⁴(0.5)⁰x⁰ + 4(1)³(0.5)¹x¹ + 6(1)²(0.5)²x² + 4(1)¹(0.5)³x³ + 1(1)⁰(0.5)⁴x⁴ First we raise 1 to the power of 4, therefore 0.5 is raised to the power of 0. For the next term, the power of 1 decreases by 1 and the power of 0.5 is increased by 1, so that the sum of the terms still equates to 4. This is done until we get 5 terms in total. Step 3. The expression is simplified as followed: = 1 = 4(1/2)x + 6(1/4)x²+ 4(1/8)x³ + (1/16)x⁴ = 1 + x + 3/2x² + 1/2x³ + 1/16x⁴