A cube B cube formula may be considered as a cube plus b cube formula or a cube minus b cube formula. These two formulas are used to expand the binomial expressions and evaluate the numerical expressions without actually calculating the cubes. Thus, a cube b cube formula is the most commonly applied formula while simplifying the numerical problems in maths. Let’s learn the formulas for both a3 + b3 and a3 – b3 here along with proofs and examples.
A Cube Plus B Cube Formula
a3 + b3 formula is used to find the sum of the cube of two numbers. The formula of a cube plus b cube is given by:
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a3 + b3 = (a + b)(a2 – ab + b2) |
Go through the proof of a3 + b3 formula given below:
A Cube + B Cube Formula Proof
To prove: a3 + b3 = (a + b)(a2 – ab + b2)
Consider LHS = a3 + b3
RHS = (a + b)(a2 – ab + b2)
The difference of numbers a3 and b3 formula can be verified by taking the RHS.
RHS = (a + b)(a2 – ab + b2)
= = a (a2 – ab + b2) + b(a2 – ab + b2)
= a3 – a2b + ab2 + a2b – ab2 + b3
= a3 + b3
= LHS
Therefore, a3 + b3 = (a + b)(a2 – ab + b2)
Hence proved.
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A Cube Minus B Cube Formula
a3 – b3 formula is used to calculate the difference of cube of two numbers. The formula of a cube minus b cube is given by:
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a3 – b3 = (a – b)(a2 + ab + b2) |
a^3 – b^3 Formula Proof
To prove: a3 – b3 = (a – b)(a2 + ab + b2)
Consider LHS = a3 – b3
RHS = (a – b)(a2 + ab + b2)
The difference of numbers a3 and b3 formula can be verified by taking the RHS.
RHS = (a – b)(a2 + ab + b2)
= = a (a2 + ab + b2) – b(a2 + ab + b2)
= a3 + a2b + ab2 – a2b – ab2 – b3
= a3 – b3
= LHS
Therefore, a3 – b3 = (a – b)(a2 + ab + b2)
Hence proved.
Solved Examples
Example 1:
Evaluate 943 + 63 using the formula of a3 plus b3.
Solution:
943 + 63
This is of the form a3 + b3, where a = 94 and b = 6.
Using the formula a3 + b3 = (a + b)(a2 – ab + b2),
943 + 63 = (94 + 6) [(94)2 – (94)(6) + (6)2]
= 100(8836 – 564 + 36)
= 100(8308)
= 830800
Example 2:
Simplify: 253 – 125
Solution:
253 – 125 = 253 – 53
This is of the form a3 – b3, where a = 25 and b = 5.
Using the formula a3 – b3 = (a – b)(a2 + ab + b2),
253 – 53 = (25 – 5) [(25)2 + (25)(5) + (5)2]
= 20(625 + 125 + 25)
= 20(775)
= 15500
Therefore, 253 – 125 = 15500.
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