Which expression is a sum of cubes

Q: Which expression is a sum of cubes?

A polynomial in the form a 3 + b 3 is called a sum of cubes.

Consider a^{9}+27b^{12}. Rewrite a^{9}+27b^{12} as \left(a^{3}\right)^{3}+\left(3b^{4}\right)^{3}. The sum of cubes can be factored using the rule: p^{3}+q^{3}=\left(p+q\right)\left(p^{2}-pq+q^{2}\right).

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Step by Step Solution
Step 1 :
Step 2 :
Step 3 :
Final result :
Which expression is a sum of cubes?
Which expression is a difference of cubes?
Which expression is a perfect cube?
What is the cube root of b27?

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))) +  53b18
 Step  2  :

Equation at the end of step 2 :

  (0 -  (23•3a15)) +  53b18

Step 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

Final result :

  125b18 - 24a15

Which expression is a sum of cubes?

A polynomial in the form a ³ + b ³ is called a sum of cubes.

Which expression is a difference of cubes?

A polynomial in the form a 3 – b 3 is called a difference of cubes.

Which expression is a perfect cube?

The perfect cube formula is simply expressed as some integer x to the power or exponent of 3; i.e., x3 which is equal to x⋅x⋅x x ⋅ x ⋅ x . This formula also works with negative numbers because the exponent of 3 is an odd number.