Mathematics brings a mixture of emotions to us. To some, it is absolutely rewarding to get a mathematical problem solved while to some, it spells dread. But there is no way we can avoid calculations in our daily life. From choosing the best and shortest route to the destination to calculating the amount of discount you can avail on your grocery- everything is all about mathematics.
Just as Unitary Method, Time and Distance, Percentage, etc covered by arithmetic are a wing of mathematics, so is algebra, calculus, geometry, and trigonometry. In each wing, there are particular sets of formulae that one needs to memorise. There is not one, not two formulas in mathematics but a whole long list of it! A few examples of mathematical formulas would be the binary formula, calculus formulas, the difference of cubes formula, the difference of squares formula, and so on and so forth. \[a^{3}\] - \[b^{3}\] is an elementary level algebraic formula.
Algebra is a combination of both numerical and letters. Both represent the unknown quantity of the practical situation where a formula is applied to. When the numbers and letters come together with the factorials and matrices, formulas are formed. Algebraic formulas are put to use to particular situations for solving them. One of the most widely used formulas in maths is the formula of \[a^{3}\] - \[b^{3}\]. Various kinds of problems can be solved using this formula.
\[a^{3}\] - \[b^{3}\]
The formula of \[a^{3}\] - \[b^{3}\] is as follows:
(\[a^{3}\] - \[b^{3}\]) = (a-b) (\[a^{2}\] + \[b^{2}\] + ab)
Let us consider that a=8 and b=6,
Then, if the value is put to the formula, the equation will be:
(\[a^{3}\] - \[b^{3}\]) = (8-6) (\[8^{2}\] +\[6^{2}\]+ 8 x 6)
Or, (512 - 216) = 2 (64 + 36 + 48)
Or, (512 - 216) = 2 x 148
So, (a-b) (a2 + b2 + ab) = 296.
This formula is somewhat similar to the formula of a3 – b3 only with the change in the addition sign. This formula stands for a special identity instead of being a proper formula. Remembering any one helps in remembering the other.
(\[a^{3}\] - \[b^{3}\]) = (a + b) (\[a^{2}\] – ab + \[b^{2}\])
This particular formula is derived from a simpler form of (a + b)3. The derivation of this formula is as follows:
\[(a + b)^{3}\]) = \[a^{3}\] + \[b^{3}\] + 3ab (a + b)
Or, \[a^{3}\] + \[b^{3}\] = (a + b)3 – 3ab (a + b)
Or, \[a^{3}\] + \[b^{3}\] = (a + b) [\[(a+b)^{2}\]– 3ab]
Or, \[a^{3}\] + \[b^{3}\] = (a + b) (\[a^{2}\] + 2ab + \[b^{2}\] – 3ab)
So, \[a^{3}\] + \[b^{3}\] = (a + b) (\[a^{2}\] – ab + \[b^{2}\])
Let us consider, a= 5 and b= 7.
Then, if the value is put to the formula, the equation will be:
(\[5^{3}\] + \[7^{3}\]) = (5 + 7) (\[5^{2}\] + \[7^{2}\] – 5*7)
Or, (125 + 343) = 12 (\[5^{2}\] – 35 + \[7^{2}\])
Or, (125 + 343) = 12 (25 – 35 + 49)
Or, (125 + 343) = 12 x 39
So, (\[a^{2}\] – ab + \[b^{2}\]) = 468
A number of real-life problems are solved with the help of algebraic formulas. The application of them is widely spread across basic and advanced mathematics. Learning these formulas help in understanding more complex formulas as one comes across in advanced or higher levels of studies. There is no denying that Maths is the mother of all subjects. Thus the formula one learns here forms the basis of learning other subjects too. From geography to economics or statistics to Physics, mathematical formulae are indispensable. A formula like \[a^{3}\] - \[b^{3}\] always turns out as useful to solve even complex problems of higher levels of difficulty.
It is a must for students to learn all the algebraic formulae by heart with special emphasis on basic ones like \[a^{3}\] - \[b^{3}\]. This will enable them to solve any mathematical problems with ultimate mastery. With the foundation strong, the solving of advanced-level problems will become easy for them. But, one thing must be kept in mind that memorising these formulae without understanding the derivation and applications will not be a fruitful idea. Practice makes a man perfect- it is the ap test saying applicable to mathematics and one should always abide by that.
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