Simplify: 2u+2v/u+v+3(4u^2+4v^2/u+v-3)(u^2+2uv+v^2-9/16u^4-16v^4) Tiger Algebra Solver (2024)

Step 1 :

Equation at the end of step 1 :

 v (v²) 9 ((2u+(2•—))+v)+((3•((((4•(u²))+(4•————))+v)-3))•(((((u²)+2uv)+(v²))-(——•(u⁴)))-2⁴v⁴)) u u 16 Step 2 :
 9 Simplify —— 16

Equation at the end of step 2 :

 v (v²) 9 ((2u+(2•—))+v)+((3•((((4•(u²))+(4•————))+v)-3))•(((((u²)+2uv)+(v²))-(——•u⁴))-2⁴v⁴)) u u 16

Step 3 :

Equation at the end of step 3 :

 v (v²) 9u⁴ ((2u+(2•—))+v)+((3•((((4•(u²))+(4•————))+v)-3))•(((((u²)+2uv)+(v²))-———)-2⁴v⁴)) u u 16

Step 4 :

Rewriting the whole as an Equivalent Fraction :

4.1Subtracting a fraction from a whole

Rewrite the whole as a fraction using 16 as the denominator :

 u² + 2uv + v² (u² + 2uv + v²) • 16 u² + 2uv + v² = ————————————— = ———————————————————— 1 16

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

Trying to factor a multi variable polynomial :

4.2 Factoringu² + 2uv + v²

Try to factor this multi-variable trinomial using trial and errorFound a factorization:(u + v)•(u + v)

Detecting a perfect square :

4.3u²+2uv+v² is a perfect squareIt factors into (u+v)•(u+v)
which is another way of writing (u+v)²

How to recognize a perfect square trinomial: • It has three terms • Two of its terms are perfect squares themselves • The remaining term is twice the product of the square roots of the other two terms

Adding fractions that have a common denominator :

4.4 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:

 (u+v)² • 16 - (9u⁴) -9u⁴ + 16u² + 32uv + 16v² ——————————————————— = ————————————————————————— 16 16

Equation at the end of step 4 :

 v (v²) (-9u⁴+16u²+32uv+16v²) ((2u+(2•—))+v)+((3•((((4•(u²))+(4•————))+v)-3))•(—————————————————————-2⁴v⁴)) u u 16

Step 5 :

Rewriting the whole as an Equivalent Fraction :

5.1Subtracting a whole from a fraction

Rewrite the whole as a fraction using 16 as the denominator :

Checking for a perfect cube :

5.2-9u⁴ + 16u² + 32uv + 16v² is not a perfect cube

Adding fractions that have a common denominator :

5.3 Adding up the two equivalent fractions

 (-9u⁴+16u²+32uv+16v²) - (2⁴v⁴ • 16) -9u⁴ + 16u² + 32uv - 256v⁴ + 16v² ——————————————————————————————————— = ————————————————————————————————— 16 16

Equation at the end of step 5 :

 v (v²) (-9u⁴+16u²+32uv-256v⁴+16v²) ((2u+(2•—))+v)+((3•((((4•(u²))+(4•————))+v)-3))•———————————————————————————) u u 16  Step 6 :
 v² Simplify —— u

Equation at the end of step 6 :

 v v² (-9u⁴+16u²+32uv-256v⁴+16v²) ((2u+(2•—))+v)+((3•((((4•(u²))+(4•——))+v)-3))•———————————————————————————) u u 16  Step 7 :

Equation at the end of step 7 :

 v 4v² (-9u⁴+16u²+32uv-256v⁴+16v²) ((2u+(2•—))+v)+((3•(((2²u²+———)+v)-3))•———————————————————————————) u u 16

Step 8 :

Rewriting the whole as an Equivalent Fraction :

8.1Adding a fraction to a whole

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

 2²u² 2²u² • u 2²u² = ———— = ———————— 1 u

Adding fractions that have a common denominator :

8.2 Adding up the two equivalent fractions

 2²u² • u + 4v² 4u³ + 4v² —————————————— = ————————— u u

Equation at the end of step 8 :

 v (4u³+4v²) (-9u⁴+16u²+32uv-256v⁴+16v²) ((2u+(2•—))+v)+((3•((—————————+v)-3))•———————————————————————————) u u 16

Step 9 :

Rewriting the whole as an Equivalent Fraction :

9.1Adding a whole to a fraction

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

 v v • u v = — = ————— 1 u

Step 10 :

Pulling out like terms :

10.1 Pull out like factors:

4u³ + 4v²=4•(u³ + v²)

Trying to factor as a Sum of Cubes:

10.2 Factoring: u³ + v²

Theory:A sum 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: u³ is the cube of u¹

Check: v ² is not a cube !!
Ruling:Binomial can not be factored as the difference of two perfect cubes

Adding fractions that have a common denominator :

10.3 Adding up the two equivalent fractions

 4 • (u³+v²) + v • u 4u³ + uv + 4v² ——————————————————— = —————————————— u u

Equation at the end of step 10 :

 v (4u³+uv+4v²) (-9u⁴+16u²+32uv-256v⁴+16v²) ((2u+(2•—))+v)+((3•(————————————-3))•———————————————————————————) u u 16

Step 11 :

Rewriting the whole as an Equivalent Fraction :

11.1Subtracting a whole from a fraction

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

 3 3 • u 3 = — = ————— 1 u

Trying to factor a multi variable polynomial :

11.2 Factoring4u³ + uv + 4v²

Try to factor this multi-variable trinomial using trial and errorFactorization fails

Adding fractions that have a common denominator :

11.3 Adding up the two equivalent fractions

 (4u³+uv+4v²) - (3 • u) 4u³ + uv - 3u + 4v² —————————————————————— = ——————————————————— u u

Equation at the end of step 11 :

 v (4u³+uv-3u+4v²) (-9u⁴+16u²+32uv-256v⁴+16v²) ((2u+(2•—))+v)+((3•———————————————)•———————————————————————————) u u 16

Step 12 :

Checking for a perfect cube :

12.14u³+uv-3u+4v² is not a perfect cube

Equation at the end of step 12 :

 v 3•(4u³+uv-3u+4v²) (-9u⁴+16u²+32uv-256v⁴+16v²) ((2u+(2•—))+v)+(—————————————————•———————————————————————————) u u 16

Step 13 :

Equation at the end of step 13 :

 v 3•(4u³+uv-3u+4v²)•(-9u⁴+16u²+32uv-256v⁴+16v²) ((2u+(2•—))+v)+————————————————————————————————————————————— u 16u

Step 14 :

 v Simplify — u

Equation at the end of step 14 :

 v 3•(4u³+uv-3u+4v²)•(-9u⁴+16u²+32uv-256v⁴+16v²) ((2u+(2•—))+v)+————————————————————————————————————————————— u 16u

Step 15 :

Rewriting the whole as an Equivalent Fraction :

15.1Adding a fraction to a whole

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

 2u 2u • u 2u = —— = —————— 1 u

Adding fractions that have a common denominator :

15.2 Adding up the two equivalent fractions

 2u • u + 2v 2u² + 2v ——————————— = ———————— u u

Equation at the end of step 15 :

 (2u²+2v) 3•(4u³+uv-3u+4v²)•(-9u⁴+16u²+32uv-256v⁴+16v²) (————————+v)+————————————————————————————————————————————— u 16u

Step 16 :

Rewriting the whole as an Equivalent Fraction :

16.1Adding a whole to a fraction

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

 v v • u v = — = ————— 1 u

Step 17 :

Pulling out like terms :

17.1 Pull out like factors:

2u² + 2v=2•(u² + v)

Adding fractions that have a common denominator :

17.2 Adding up the two equivalent fractions

 2 • (u²+v) + v • u 2u² + uv + 2v —————————————————— = ————————————— u u

Equation at the end of step 17 :

 (2u²+uv+2v) 3•(4u³+uv-3u+4v²)•(-9u⁴+16u²+32uv-256v⁴+16v²) ———————————+————————————————————————————————————————————— u 16u

Step 18 :

Trying to factor a multi variable polynomial :

18.1 Factoring2u² + uv + 2v

Try to factor this multi-variable trinomial using trial and errorFactorization fails

Calculating the Least Common Multiple :

18.2 Find the Least Common Multiple

The left denominator is : u

The right denominator is : 16u

Number of times each prime factor
appears in the factorization of:
Prime Factor	Left Denominator	Right Denominator	L.C.M = Max {Left,Right}
2	0	4	4
Product of all Prime Factors	1	16	16

Number of times each Algebraic Factor
appears in the factorization of:
Algebraic Factor	Left Denominator	Right Denominator	L.C.M = Max {Left,Right}
u	1	1	1

Least Common Multiple:
16u

Calculating Multipliers :

18.3 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=16

Right_M=L.C.M/R_Deno=1

Making Equivalent Fractions :

18.4 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. (2u²+uv+2v) • 16 —————————————————— = ———————————————— L.C.M 16u  R. Mult. • R. Num. 3 • (4u³+uv-3u+4v²) • (-9u⁴+16u²+32uv-256v⁴+16v²) —————————————————— = ————————————————————————————————————————————————— L.C.M 16u

Adding fractions that have a common denominator :

18.5 Adding up the two equivalent fractions

 (2u²+uv+2v) • 16 + 3 • (4u³+uv-3u+4v²) • (-9u⁴+16u²+32uv-256v⁴+16v²) -108u⁷-27u⁵v+273u⁵-108u⁴v²+384u⁴v-3072u³v⁴+192u³v²+48u³v-144u³+288u²v²-288u²v+32u²-768uv⁵+2304uv⁴+432uv³-144uv²+16uv-3072v⁶+192v⁴+32v ———————————————————————————————————————————————————————————————————— = ————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————— 16u 16u

Final result :

 -108u⁷-27u⁵v+273u⁵-108u⁴v²+384u⁴v-3072u³v⁴+192u³v²+48u³v-144u³+288u²v²-288u²v+32u²-768uv⁵+2304uv⁴+432uv³-144uv²+16uv-3072v⁶+192v⁴+32v ————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————— 16u

Simplify: 2u+2v/u+v+3(4u^2+4v^2/u+v-3)(u^2+2uv+v^2-9/16u^4-16v^4) Tiger Algebra Solver (2024)

Step 1 :

Equation at the end of step 1 :

Step 2 :

Equation at the end of step 2 :

Step 3 :

Equation at the end of step 3 :

Step 4 :

Rewriting the whole as an Equivalent Fraction :

Trying to factor a multi variable polynomial :

Detecting a perfect square :

Adding fractions that have a common denominator :

Equation at the end of step 4 :

Step 5 :

Rewriting the whole as an Equivalent Fraction :

Checking for a perfect cube :

Adding fractions that have a common denominator :

Equation at the end of step 5 :

Step 6 :

Equation at the end of step 6 :

Step 7 :

Equation at the end of step 7 :

Step 8 :

Rewriting the whole as an Equivalent Fraction :

Adding fractions that have a common denominator :

Equation at the end of step 8 :

Step 9 :

Rewriting the whole as an Equivalent Fraction :

Step 10 :

Pulling out like terms :

Trying to factor as a Sum of Cubes:

Adding fractions that have a common denominator :

Equation at the end of step 10 :

Step 11 :

Rewriting the whole as an Equivalent Fraction :

Trying to factor a multi variable polynomial :

Adding fractions that have a common denominator :

Equation at the end of step 11 :

Step 12 :

Checking for a perfect cube :

Equation at the end of step 12 :

Step 13 :

Equation at the end of step 13 :

Step 14 :

Equation at the end of step 14 :

Step 15 :

Rewriting the whole as an Equivalent Fraction :

Adding fractions that have a common denominator :

Equation at the end of step 15 :

Step 16 :

Rewriting the whole as an Equivalent Fraction :

Step 17 :

Pulling out like terms :

Adding fractions that have a common denominator :

Equation at the end of step 17 :

Step 18 :

Trying to factor a multi variable polynomial :

Calculating the Least Common Multiple :

Calculating Multipliers :

Making Equivalent Fractions :

Adding fractions that have a common denominator :

Final result :

References