Question

If \[a - b = 6\], and \[ab = 20\], find the value of \[{a^3} - {b^3}\]

Answer 1

Hint:
Here we will use suitable algebraic identity which includes the given expression. We will then substitute the given values in the algebraic identity. We will simplify it further to get the required answer.

Complete step by step solution:
We have to find the value of \[{a^3} - {b^3}\].
As we can see the expression is the difference of cube of two terms, so we will use suitable identity such that the expression is there in the identity.
We will use the identity:
\[{\left( {a - b} \right)^3} = {a^3} - 3ab\left( {a - b} \right) - {b^3}\]…….\[\left( 1 \right)\]
Substituting \[a - b = 6\] and \[ab = 20\] in equation \[\left( 1 \right)\], we get
\[ \Rightarrow {\left( 6 \right)^3} = {a^3} - 3 \times 20 \times 6 - {b^3}\]
Simplifying the equation, we get
\[ \Rightarrow 216 = {a^3} - 360 - {b^3}\]
Adding 360 on both the sides, we get
\[\begin{array}{l}{a^3} - {b^3} = 216 + 360\\{a^3} - {b^3} = 576\end{array}\]

Therefore, we got the value of \[{a^3} - {b^3}\] as 576.

Additional information:
Algebraic identities are the algebraic equations which are valid for all values of the variables. All the algebraic identities are derived from binomial theorems. We can verify the algebraic identities by using a substitution method in which we substitute the value of the variable and perform arithmetic operations. There are many algebraic identities that are used to find the values of the variables.

Note:
Algebraic expressions are an expression which is made up of constants and variables along with some algebraic operation. Algebraic expression is different from algebraic equations as in the former there is no equal to sign. There are three types of algebraic expression which are Monomial expression, Binomial expression and Polynomial expression. Monomial expression has only 1 term and binomial expression has 2 terms. We can derive algebraic expressions by using arithmetic operations such as \[\left( { + , - , \times , \div } \right)\].