X 3 + y 3 + z 3 = 33 has a solution in z

x3/y3+y3/z3+z3/x3-3

This deals with adding, subtracting and finding the least comtháng multiple.

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Step by Step Solution

Reformatting the đầu vào :

Changes made lớn your input đầu vào should not affect the solution: (1): "x3" was replaced by "x^3". 5 more similar replacement(s).

Step 1 :

z3 Simplify —— x3Equation at the over of step 1 : (x3) (y3) z3 ((————+————)+——)-3 (y3) (z3) x3

Step 2 :

y3 Simplify —— z3Equation at the kết thúc of step 2 : (x3) y3 z3 ((————+——)+——)-3 (y3) z3 x3

Step 3 :

x3 Simplify —— y3Equation at the over of step 3 : x3 y3 z3 ((—— + ——) + ——) - 3 y3 z3 x3

Step 4 :

Calculating the Least Common Multiple :4.1 Find the Least Common Multiple The left denominator is : y3 The right denominator is : z3

Number of times each Algebraic Factorappears in the factorization of:AlgebraicFactorLeftDenominatorRightDenominatorL.C.M = MaxLeft,Right

y	3	0	3
z	0	3	3

Least Common Multiple: y3z3

Calculating Multipliers :

4.2 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_DenoLeft_M=L.C.M/L_Deno=z3Right_M=L.C.M/R_Deno=y3

Making Equivalent Fractions :

4.3 Rewrite the two fractions into lớn equivalent fractionsTwo fractions are called equivalent if they have sầu the same numeric value. For example : một nửa and 2/4 are equivalent, y/(y+1)2 & (y2+y)/(y+1)3 are equivalent as well. To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective sầu Multiplier.

L. Mult. • L. Num. x3 • z3 —————————————————— = ——————— L.C.M y3z3 R. Mult. • R.

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Num. y3 • y3 —————————————————— = ——————— L.C.M y3z3 Adding fractions that have sầu a comtháng denominator :4.4 Adding up the two equivalent fractions Add the two equivalent fractions which now have a common denominatorCombine the numerators together, put the sum or difference over the comtháng denominator then reduce khổng lồ lowest terms if possible:

x3 • z3 + y3 • y3 x3z3 + y6 ————————————————— = ————————— y3z3 y3z3 Equation at the end of step 4 : (x3z3 + y6) z3 (——————————— + ——) - 3 y3z3 x3

Step 5 :

Trying khổng lồ factor as a Sum of Cubes:5.1 Factoring: x3z3+y6 Theory:A sum of two perfect cubes, a3+b3 can be factored inkhổng lồ :(a+b)•(a2-ab+b2)Proof: (a+b)•(a2-ab+b2) = a3-a2b+ab2+ba2-b2a+b3=a3+(a2b-ba2)+(ab2-b2a)+b3=a3+0+0+b3=a3+b3Check: x3 is the cube of x1Check: z3 is the cube of z1Check: y6 is the cube of y2Factorization is :(xz + y2)•(x2z2 - xy2z + y4)

Trying khổng lồ factor a multi variable polynomial :5.2 Factoringx2z2 - xy2z + y4Try to lớn factor this multi-variable trinomial using trial và errorFactorization fails

Calculating the Least Common Multiple :5.3 Find the Least Common Multiple The left denominator is : y3z3 The right denominator is : x3

Number of times each Algebraic Factorappears in the factorization of:AlgebraicFactorLeftDenominatorRightDenominatorL.C.M = MaxLeft,Right

x	0	3	3
y	3	0	3
z	3	0	3

Least Common Multiple: x3y3z3

Calculating Multipliers :

5.4 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_DenoLeft_M=L.C.M/L_Deno=x3Right_M=L.C.M/R_Deno=y3z3

Making Equivalent Fractions :

5.5 Rewrite the two fractions into equivalent fractions

L. Mult. • L. Num. (xz+y2) • (x2z2-xy2z+y4) • x3 —————————————————— = ————————————————————————————— L.C.M x3y3z3 R. Mult. • R. Num. z3 • y3z3 —————————————————— = ————————— L.C.M x3y3z3 Adding fractions that have sầu a common denominator :5.6 Adding up the two equivalent fractions

(xz+y2) • (x2z2-xy2z+y4) • x3 + z3 • y3z3 x6z3 + x3y6 + y3z6 ————————————————————————————————————————— = —————————————————— x3y3z3 x3y3z3 Equation at the over of step 5 : (x6z3 + x3y6 + y3z6) ———————————————————— - 3 x3y3z3

Step 6 :

Rewriting the whole as an Equivalent Fraction :6.1Subtracting a whole from a fraction Rewrite the whole as a fraction using x3y3z3 as the denominator :

3 3 • x3y3z3 3 = — = —————————— 1 x3y3z3 Equivalent fraction : The fraction thus generated looks different but has the same value as the whole Common denominator : The equivalent fraction và the other fraction involved in the calculation mô tả the same denominator

Trying khổng lồ factor a multi variable polynomial :6.2 Factoringx6z3 + x3y6 + y3z6Try khổng lồ factor this multi-variable trinomial using trial và errorFactorization fails

Adding fractions that have sầu a common denominator :6.3 Adding up the two equivalent fractions

(x6z3+x3y6+y3z6) - (3 • x3y3z3) x6z3 + x3y6 - 3x3y3z3 + y3z6 ——————————————————————————————— = ———————————————————————————— x3y3z3 x3y3z3 Checking for a perfect cube :6.4x6z3 + x3y6 + 3x3y3z3 + y3z6 is not a perfect cube