Math
Cube Root Formula  Difference of Cubes & Sum of Cubes
Cube Root Formula
Before we look at the actual sum and differences of cube formula, you first need to know cube Formulas are necessary to study. This is a part of simple mathematics itself and learned during early school days. It is commonly used for complex calculations where cubes are given or problem is stated in the form of cubic equations.
You should know how to break the cubic equations and continue with solving the difficult problems. With a list of basic cubic formulas, you just have to put the values and solve any particular problem. Here, we will discuss how to calculate the sum of cubes and the difference of cubes.
\[\ 5\;cube = 5^{3} = 125\]
\[\ Cube\;Root\;of\;125 = \sqrt[3]{125} = 5 \]
Cubes of some number are as follows –
\[\ 1^{3} = 1\]
\[\ 2^{3} = 8\]
\[\ 3^{3} = 27\]
\[\ 4^{3} = 64\]
\[\ 5^{3} = 125\]
\[\ 6^{3} = 216\]
\[\ 7^{3} = 343\]
\[\ 8^{3} = 512\]
\[\ 9^{3} = 729\]
\[\ 10^{3} = 1000\]
The Difference of Cubes Formula
Factoring the two cubes is almost identical with a simple difference of minus sign. You should know where to use minus sign in the equation and solve the problem in minutes. Sometimes, you have to go through Brute Force technique to get through typical mathematical problems. But it takes a lot of time and efforts in recognizing these types of problems where the standardized formula is required. Once you will know the formula for factoring a^{3 }+ b^{3} or a^{3} – b^{3, }then solving equations is easier as substituting the values in the formula.
\[\LARGE a^{3}b^{3}=(a+b)(a^{2}+ab+b^{2})\]
Sum of Cubes Formula
The other common factoring formula that you should know is very much similar to the earlier one with a single difference of sign. Here, is a quick representation of how the sum of cubes formula can be given in mathematics.
Look at the formula given carefully. When you will multiply the righthand side then you will get the lefthand side equation. Also, keep in mind that cube equation cannot be factorized further.
\[\large a^{3}+b^{3}=\left(a+b\right)\left(a^{2}ab+b^{2}\right)\]
You just need the right memorization skills in terms of two factorizations so that minus sign can be utilized wisely. Rest of the factors are almost the same if you are sure of the location of that minus sign in the equation. Few people use interesting techniques to keep track of minus sign and avoid any mistake during the final attempt that may be school exam or any other competitive test.
Important of Cube Formula for Students
This is easy to memorize formulas but actual implementation needs to be correct. Whatever method fits best for solving cubic equation, you can use the same. The best idea is to practice a number of problems before the final attempt to make sure that concept is clear and used as per expectations only.
Also, don’t waste time in factoring more because this was maximum we have done. The best idea is to learn the actual implementation and understand how they can be beneficial in the real world. The cube formulas are used in our daily life too and a major part of typical chemistry and physics derivations. They are used when you are preparing for competitive exams and higher studies. So, all the best and start learning the different cube formulas right away with right technique and approach.
Solution:
a^{3} – b^{3} = (a – b)(a^{2} + ab + b^{2})
From the given equation,
a = 5 ; b = 3
5^{3 }– 3^{3}
= (5 – 3) (5^{2} + (5)(3) + 3^{2})
= 2 X (25 + 15 + 9)
= 2 X 49
= 98
Question 2: What is a value of 5^{3 } + 3^{3} ?
Solution:
a^{3} – b^{3} = (a + b)(a^{2} – ab + b^{2})
From the given equation,
a = 5 ; b = 3
5^{3 } + 3^{3}
= (5 + 3) (5^{2} – (5)(3) + 3^{2})
= 8 X (25 – 15 + 9)
= 8 X 19
= 152

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