x^-3 -y^-3/x^-3 y^-1 +(xy)^2 +y^-3 x^-1
Answers
(x1/3-y1/3)(x2/3-x1/3y1/3+y2/3)
Final result :
(x - y) • (3x2 - xy + 3y2)
——————————————————————————
27
Step by step solution :
Step 1 :
y2
Simplify ——
3
Equation at the end of step 1 :
(x1) (y1) (x2) (x1) (y1) y2
(————-————)•((————-(————•————))+——)
3 3 3 3 3 3
Step 2 :
y
Simplify —
3
Equation at the end of step 2 :
(x1) (y1) (x2) (x1) y y2
(————-————)•((————-(————•—))+——)
3 3 3 3 3 3
Step 3 :
x
Simplify —
3
Equation at the end of step 3 :
(x1) (y1) (x2) x y y2
(————-————)•((————-(—•—))+——)
3 3 3 3 3 3
Step 4 :
x2
Simplify ——
3
Equation at the end of step 4 :
(x1) (y1) x2 xy y2
(————-————)•((——-——)+——)
3 3 3 9 3
Step 5 :
Calculating the Least Common Multiple :
5.1 Find the Least Common Multiple
The left denominator is : 3
The right denominator is : 9
Number of times each prime factor
appears in the factorization of:
Prime
Factor Left
Denominator Right
Denominator L.C.M = Max
{Left,Right}
3 1 2 2
Product of all
Prime Factors 3 9 9
Least Common Multiple:
9
Calculating Multipliers :
5.2 Calculate multipliers for the two fractions
Denote the Least Common Multiple by L.C.M
Denote the Left Multiplier by Left_M
Denote the Right Multiplier by Right_M
Denote the Left Deniminator by L_Deno
Denote the Right Multiplier by R_Deno
Left_M = L.C.M / L_Deno = 3
Right_M = L.C.M / R_Deno = 1
Making Equivalent Fractions :
5.3 Rewrite the two fractions into equivalent fractions
Two fractions are called equivalent if they have the same numeric value.
For example : 1/2 and 2/4 are equivalent, y/(y+1)2 and (y2+y)/(y+1)3 are equivalent as well.
To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.
L. Mult. • L. Num. x2 • 3
—————————————————— = ——————
L.C.M 9
R. Mult. • R. Num. xy
—————————————————— = ——
L.C.M 9
Adding fractions that have a common denominator :
5.4 Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:
x2 • 3 - (xy) 3x2 - xy
————————————— = ————————
9 9
Equation at the end of step 5 :
(x1) (y1) (3x2-xy) y2
(————-————)•(————————+——)
3 3 9 3
Step 6 :
Step 7 :
Pulling out like terms :
7.1 Pull out like factors :
3x2 - xy = x • (3x - y)
Calculating the Least Common Multiple :
7.2 Find the Least Common Multiple
The left denominator is : 9
The right denominator is : 3
Number of times each prime factor
appears in the factorization of:
Prime
Factor Left
Denominator Right
Denominator L.C.M = Max
{Left,Right}
3 2 1 2
Product of all
Prime Factors 9 3 9
Least Common Multiple:
9
Calculating Multipliers :
7.3 Calculate multipliers for the two fractions
Denote the Least Common Multiple by L.C.M
Denote the Left Multiplier by Left_M
Denote the Right Multiplier by Right_M
Denote the Left Deniminator by L_Deno
Denote the Right Multiplier by R_Deno
Left_M = L.C.M / L_Deno = 1
Right_M = L.C.M / R_Deno = 3
Making Equivalent Fractions :
7.4 Rewrite the two fractions into equivalent fractions
L. Mult. • L. Num. x • (3x-y)
—————————————————— = ——————————
L.C.M 9
R. Mult. • R. Num. y2 • 3
—————————————————— = ——————
L.C.M 9
Adding fractions that have a common denominator :
7.5 Adding up the two equivalent fractions
x • (3x-y) + y2 • 3 3x2 - xy + 3y2
——————————————————— = ——————————————
9 9
Equation at the end of step 7 :
(x1) (y1) (3x2-xy+3y2)
(————-————)•————————————
3 3 9
Step 8 :
y
Simplify —
3
Equation at the end of step 8 :
(x1) y (3x2 - xy + 3y2)
(———— - —) • ————————————————
3 3 9
Step 9 :
x
Simplify —
3
Equation at the end of step 9 :
x y (3x2 - xy + 3y2)
(— - —) • ————————————————
3 3 9
Step 10 :
Adding fractions which have a common denominator :
10.1 Adding fractions which have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:
x - (y) x - y
——————— = —————
3 3
Equation at the end of step 10 :
(x - y) (3x2 - xy + 3y2)
——————— • ————————————————
3 9
Step 11 :
Trying to factor a multi variable polynomial :
11.1 Factoring 3x2 - xy + 3y2
Try to factor this multi-variable trinomial using trial and error
Factorization fails
Final result :
(x - y) • (3x2 - xy + 3y2)
——————————————————————————
27