Question

If a – b = $6$ and ${{a}^{2}}+{{b}^{2}}=42$, find the value of ab.

Answer 1

Hint: The identities related to squares addition, subtraction, multiplication and division of variables have to be applied to find the value of required expression.
The identities mainly used have the square or cubic powers.

Complete step by step answer:
The identities relating variables x and y are:
${{(x+y)}^{2}}={{x}^{2}}+{{y}^{2}}+2xy$
${{(x-y)}^{2}}={{x}^{2}}+{{y}^{2}}-2xy$
${{(x+y)}^{3}}={{x}^{3}}+{{y}^{3}}+\{3xy(x+y)\}$
${{(x-y)}^{3}}={{x}^{3}}-{{y}^{3}}-\{3xy(x-y)\}$
${{(x)}^{3}}+{{(y)}^{3}}=(x+y)({{x}^{^{2}}}+{{y}^{2}}-xy)$
${{(x)}^{3}}-{{(y)}^{3}}=(x-y)({{x}^{^{2}}}+{{y}^{2}}+xy)$
The base “x” and “y” can be a whole number or a rational number and the same applies to power also.
Similarly, bases and powers can be negative or positive. This indicates that both bases and powers belong to rational numbers as rational numbers include all types of integers, zero and both positive and negative fractions.
As per first equation, a – b = $6$. Putting the value of (a – b) in the second identity listed above helps in finding the product of ab. The value of ab determined as:
$\begin{align}
  & {{(a-b)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab \\
 & {{(6)}^{2}}=42-2ab \\
 & 2ab=42-36 \\
 & 2ab=6 \\
 & ab=3
\end{align}$
This indicates that the value of expression ab is $3$.
The value of ab equal to $3$ satisfies the equation related to the square of difference of variables “a” and “b” with the sum of square of variable a, square of variable b and twice the product of variables a and b.
The identities help in reducing steps in a calculation and help in finding answers in an easy and time saving manner.

Note: The identities should be applied by finding their suitability with the conditions given.
The expressions related to identities can be identified by knowing the process of deriving the identities or by learning the identities in such a way that they are on our tips.