This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here is the solution to the problem. Simplify (2 - (y^5)/(4)y^(1)/(4) - y^-(1)/(4)). Step 1: Simplify the denominator of the fraction. The denominator is y^(1)/(4) - y^-(1)/(4). Rewrite y^-(1)/(4) as (1)/(y^1)4: y^(1)/(4) - (1)/(y^1)4 Find a common denominator, which is y^(1)/(4): = (y^1)/(4) · y^(1)/(4)y^(1)/(4) - (1)/(y^1)4 = (y^1)/(4) + (1)/(4) - 1y^(1)/(4) = (y^2)/(4) - 1y^(1)/(4) = (y^1)/(2) - 1y^(1)/(4) Step 2: Substitute the simplified denominator back into the fraction and simplify the fraction. The fraction is (y^5)/(4)denominator: (y^5)/(4)(y^1)/(2) - 1y^(1)/(4) Multiply the numerator by the reciprocal of the denominator: = y^(5)/(4) × (y^1)/(4)y^(1)/(2) - 1 Combine the y terms in the numerator using the exponent rule a^m × a^n = a^m+n: = (y^5)/(4) + (1)/(4)y^(1)/(2) - 1 = (y^6)/(4)y^(1)/(2) - 1 Simplify the exponent in the numerator: (6)/(4) = (3)/(2). = (y^3)/(2)y^(1)/(2) - 1 Step 3: Substitute this simplified fraction back into the original expression and combine terms. The original expression is 2 - fraction: 2 - (y^3)/(2)y^(1)/(2) - 1 Find a common denominator, which is (y^(1)/(2) - 1): = (2(y^1)/(2) - 1)y^(1)/(2) - 1 - (y^3)/(2)y^(1)/(2) - 1 = (2(y^1)/(2) - 1) - y^(3)/(2)y^(1)/(2) - 1 Distribute the 2 in the numerator: = (2y^1)/(2) - 2 - y^(3)/(2)y^(1)/(2) - 1 Rearrange the terms in the numerator: = (-y^3)/(2) + 2y^(1)/(2) - 2y^(1)/(2) - 1 To make the leading term in the numerator positive, multiply the numerator and denominator by -1: = (-(y^3)/(2) - 2y^(1)/(2) + 2)-(1 - y^(1)/(2)) = (y^3)/(2) - 2y^(1)/(2) + 21 - y^(1)/(2) The simplified expression is (y^3)/(2) - 2y^(1)/(2) + 21 - y^(1)/(2).