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This solution deals with simplification or other simple results.
Step by Step Solution
Step 1 :
Equation at the end of step 1 :
(0 - (24 • (a15))) + 53b18Step 2 :
Equation at the end of step 2 :
(0 - (23•3a15)) + 53b18Step 3 :
Trying to factor as a Difference of Squares :
3.1 Factoring: 125b18-24a15
Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)
Proof : (A+B) • (A-B) =
A2 - AB + BA - B2 =
A2 - AB + AB - B2 =
A2 - B2
Note : AB = BA is the commutative property of multiplication.
Note : - AB + AB equals zero and is therefore eliminated from the expression.
Check : 125 is not a square !!
Ruling : Binomial can not be factored as the
difference of two perfect squares
Trying to factor as a Difference of Cubes:
3.2 Factoring: 125b18-24a15
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 : 125 is the cube of 5
Check : 24 is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes