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Answer 1

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Step by Step Solution:

STEP 1 :

3

Simplify —

2

Equation at the end of step 1 :

1 3 3

(((——•(x3))-(8•(y3)))+((——•(x2))•y))-((—•x)•y2)

64 16 2

STEP 2 : Equation at the end of step 2

1 3 3xy2

(((——•(x3))-(8•(y3)))+((——•(x2))•y))-————

64 16 2

STEP 3 :

3

Simplify ——

16

Equation at the end of step 3 :

1 3 3xy2

(((——•(x3))-(8•(y3)))+((——•x2)•y))-————

64 16 2

STEP 4 : Equation at the end of step 4

1 3x2 3xy2

(((——•(x3))-(8•(y3)))+(———•y))-————

64 16 2

STEP 5 :

Equation at the end of step 5 :

1 3x2y 3xy2

(((——•(x3))-23y3)+————)-————

64 16 2

STEP 6 :

1

Simplify ——

64

Equation at the end of step 6 :

1 3x2y 3xy2

(((—— • x3) - 23y3) + ————) - ————

64 16 2

STEP 7 : Equation at the end of step 7

x3 3x2y 3xy2

((—— - 23y3) + ————) - ————

64 16 2

STEP 8 : Rewriting the whole as an Equivalent Fraction

8.1 Subtracting a whole from a fraction

Rewrite the whole as a fraction using 64 as the denominator :

23y3 23y3 • 64

23y3 = ———— = —————————

1 64

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

8.2 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:

x3 - (23y3 • 64) x3 - 512y3

———————————————— = ——————————

64 64

Equation at the end of step 8 :

(x3 - 512y3) 3x2y 3xy2

(———————————— + ————) - ————

64 16 2

STEP 9 : Trying to factor as a Difference of Cubes

9.1 Factoring: x3-512y3

Theory : A difference of two perfect cubes, a3 - b3 can be factored into

(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+b3

Check : 512 is the cube of 8

Check : x3 is the cube of x1

Check : y3 is the cube of y1

Factorization is :

(x - 8y) • (x2 + 8xy + 64y2)

Trying to factor a multi variable polynomial :

9.2 Factoring x2 + 8xy + 64y2

Try to factor this multi-variable trinomial using trial and error

Factorization fails

Calculating the Least Common Multiple :

9.3 Find the Least Common Multiple

The left denominator is : 64

The right denominator is : 16

Number of times each prime factor

appears in the factorization of:

Prime

Factor Left

Denominator Right

Denominator L.C.M = Max

{Left,Right}

2 6 4 6

Product of all

Prime Factors 64 16 64

Least Common Multiple:

64

Calculating Multipliers :

9.4 Calculate multipliers for the two fractions

Left_M = L.C.M / L_Deno = 1

Right_M = L.C.M / R_Deno = 4

Making Equivalent Fractions :

9.5 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. (x-8y) • (x2+8xy+64y2)

—————————————————— = ——————————————————————

L.C.M 64

R. Mult. • R. Num. 3x2y • 4

—————————————————— = ————————

L.C.M 64

Adding fractions that have a common denominator :

9.6 Adding up the two equivalent fractions

(x-8y) • (x2+8xy+64y2) + 3x2y • 4 x3 + 12x2y - 512y3

————————————————————————————————— = ——————————————————

64 64

Equation at the end of step 9 :

(x3 + 12x2y - 512y3) 3xy2

———————————————————— - ————

64 2

STEP 10 : Trying to factor a multi variable polynomial

10.1 Factoring x3 + 12x2y - 512y3

Try to factor this multi-variable trinomial using trial and error

Factorization fails

Calculating the Least Common Multiple :

10.2 Find the Least Common Multiple

The left denominator is : 64

The right denominator is : 2

Number of times each prime factor

appears in the factorization of:

Prime

Factor Left

Denominator Right

Denominator L.C.M = Max

{Left,Right}

2 6 1 6

Product of all

Prime Factors 64 2 64

Least Common Multiple:

64

Calculating Multipliers :

10.3 Calculate multipliers for the two fractions

Left_M = L.C.M / L_Deno = 1

Right_M = L.C.M / R_Deno = 32

Making Equivalent Fractions :

10.4 Rewrite the two fractions into equivalent fractions

L. Mult. • L. Num. (x3+12x2y-512y3)

—————————————————— = ————————————————

L.C.M 64

R. Mult. • R. Num. 3xy2 • 32

—————————————————— = —————————

L.C.M 64

Adding fractions that have a common denominator :

10.5 Adding up the two equivalent fractions

(x3+12x2y-512y3) - (3xy2 • 32) x3 + 12x2y - 96xy2 - 512y3

—————————————————————————————— = ——————————————————————————

64 64

Checking for a perfect cube :

10.6 x3 + 12x2y - 96xy2 - 512y3 is not a perfect cube

Final result :

x3 + 12x2y - 96xy2 - 512y3

——————————————————————————

64

Answer 2

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