Question

Find the squares of 99 using the identity ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$ ?

Answer 1

Hint: We need to find the square of the number 99. Firstly, we write the given number 99 as the difference between 100 and 1. Then, we use the formula of ${{\left( a-b \right)}^{2}}$ in mathematics to get the square of the number 99.

Complete step by step solution:
We are given a number and need to find out the value of ${{99}^{2}}$ . We will be solving the given question by writing the number 99 as the difference between 100 and 1 and then evaluating the expression using the formula of ${{\left( a-b \right)}^{2}}$ .
The square of a number is defined as the result of multiplying the number by itself. The square of a number $n$ is given by $n\times n$ also written as ${{n}^{2}}$ .
Let us now understand how to find the square of a given number through an example.
Example:
What is 5 squared?
The square of the number 5 is obtained by multiplying the number 5 with itself.
Applying the same, we get,
$\Rightarrow 5\times 5$
The product of number with itself that is $n\times n$ can be also written as ${{n}^{2}}$ .
Writing the same, we get,
$\Rightarrow {{5}^{2}}$
The result of the above expression is $25$
Substituting the same, we get,
$\therefore {{5}^{2}}=25$
According to our question, we need to find the value of ${{99}^{2}}$ using an identity.
Expressing the number 99 as the difference between 100 and 1, we get,
$\Rightarrow 99=\left( 100-1 \right)$
Squaring the above equation on both sides, we get,
$\Rightarrow {{99}^{2}}={{\left( 100-1 \right)}^{2}}$
The formula of ${{\left( a-b \right)}^{2}}$ is given by
$\Rightarrow {{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab$
Here,
a: 100
b: 1
Applying the same for the above equation, we get,
$\Rightarrow {{99}^{2}}={{100}^{2}}+{{1}^{2}}-\left( 2\times 100\times 1 \right)$
Simplifying the above equation, we get,
$\Rightarrow {{99}^{2}}=10000+1-\left( 200 \right)$
Let us evaluate it further.
$\Rightarrow {{99}^{2}}=10000+\left( 1-200 \right)$
$\Rightarrow {{99}^{2}}=10000-199$
$\therefore {{99}^{2}}=9801$

The value of the square of the number 99 is 9801.

Note: The result can be cross-checked by applying the square root on both sides of the equation ${{99}^{2}}=9801$ .
LHS:
$\Rightarrow \sqrt{{{\left( 99 \right)}^{2}}}$
$\Rightarrow 99$
RHS:
$\Rightarrow \sqrt{9801}$
$\Rightarrow \sqrt{{{\left( 99 \right)}^{2}}}$
$\Rightarrow 99$
LHS = RHS. The result attained is correct.