Question

Simplify the equation

Answer 1

`\sf (c^2-2c+4)/(c)`

Explanation:

Simplification of numerator:

LCM of 1 & c² is c². So find equivalent fractions with denominator c².

`\sf c + (8)/(c^2)=(c*c)/(1*c)+(8)/(c^2)`

`\sf = (c^3)/(c^2)+(8)/(c^2)\\\\\\ =(c^3+8)/(c^2)\\\\\\=(c^3+2^3)/(c^2)`

Identity: a³ + b³ = (a +b)(a²-ab+b²)

Here, a = c & b = 2

`\sf =((c+2)(c^2-c*2+2^2))/(c^2)\\\\\\=((c+2)(c^2-2c+4))/(c^2)`

Simplification of denominator:

LCM of 1 and c is c. So, find equivalent fraction with denominator c.

`\sf 1 +(2)/(c)=(1*c)/(1*c)+(2)/(c)\\`

`\sf =(c)/(c)+(2)/(c)\\\\=(c+2)/(c)`

Simplification:

Use KCF method to simplify fraction division.

K - Keep the first fraction.

C - Change the division to multiplication

F - Flip the second fraction.

`\sf (1+(8)/(c^2))/(1+(2)/(c))= \left(1 + (8)/(c^2)\right) \ / \left(1+(2)/(c)\right)`

`\sf = \left(((c+2)(c^2-2c+4))/(c^2)\right)\ / \left((c+2)/(c)\right)\\\\\\ =((c+2)(c^2-2c+4))/(c^2)*(c)/((c+2))\\\\\\\text{\bf The common factor (c +2) is cancelled}\\\\=(c^2-2c+4)/(c)`