Question

Evaluate the following:
 $ {\left( {105} \right)^2} $
 $
  A.\,\,10025 \\
  B.\,\,11025 \\
  C.\,\,11125 \\
  D.\,\,12025 \\
  $

Answer 1

Hint: To find the value of the given exponent we use algebraic identity method in this we write the base as a sum of two numbers here we take one number as $ 100 $ and other as $ 5 $ . Then substituting values in algebraic identity and simplifying it to have value of given exponent or we can say solution of required problem.

Complete step-by-step answer:
To find the value of $ {\left( {105} \right)^2} $
We first write the base of the given exponent's term as the sum of two terms.
Which is
 $ 105 = 100 + 5 $
So, given exponent becomes:
 $ {\left( {100 + 5} \right)^2} $
Now, we will used algebraic identity to find value of given exponent:
Algebraic identity:
 $ {\left( {a + b} \right)^2} = {\left( a \right)^2} + {\left( b \right)^2} + 2\left( a \right)\left( b \right) $
Taking $ a\,\,as\,\,100\,\,and\,\,b\,\,as\,\,5 $ in above algebraic formula. We have,
 $
  {\left( {100 + 5} \right)^2} = {\left( {100} \right)^2} + {\left( 5 \right)^2} + 2\left( {100} \right)\left( 5 \right) \\
   \Rightarrow {\left( {105} \right)^2} = 10000 + 25 + 1000 \\
   \Rightarrow {\left( {105} \right)^2} = 11025 \;
  $
Therefore, from above we see that the value of $ {\left( {105} \right)^2} $ is $ 10125 $ .
So, the correct answer is “ $ 10125 $ ”.

Note: There are many different ways or methods to evaluate square of any number. We can find squares of any number by direct multiplication, also by diagonal method and by using algebraic identity method which is explained in above. By this method we can find square of a number easily when the base is as a bigger number.