Question

$ {(35)^2} - {\left( {28} \right)^2} = {m^2} $ . What is the value of m?
 $
  1){\text{ 7}} \\
  {\text{2) 21}} \\
  {\text{3) 49}} \\
  {\text{4) 14}} \\
  $

Answer 1

Hint: Here apply the difference of two squares in the given equations. Use $ {a^2} - {b^2} = (a + b)(a - b) $ and use the basic mathematical operations to simplify the steps in between for the required solution.

Complete step-by-step answer:
Take the given equation –
 $ {(35)^2} - {\left( {28} \right)^2} = {m^2} $
Apply the formula for difference of two squares.
I.E. $ {a^2} - {b^2} = (a + b)(a - b) $ in the above equation –
 $ \therefore (35 + 28)(35 - 28) = {m^2} $
Simplify using the basic mathematical operations.
 $ \therefore (63)(7) = {m^2} $
Take variable as the subject –
 $ \therefore {m^2} = 441 $
Take square root on both the sides of the equation –
 $ \therefore \sqrt {{m^2}} = \sqrt {441} $
By property square and square-root cancels each other
 $ \therefore m = \sqrt {441} $
Now, find square of the right hand side of the equation –
Here, we will use factorisation method
\[
  \therefore m = \sqrt {63 \times 7} \\
  \therefore m = \sqrt {7 \times 9 \times 7} \\
  \therefore m = \sqrt {\underline {7 \times 7} \times \underline {3 \times 3} } \\
 \]
Make pair of numbers in the product form to write in the form of squares
\[\therefore m = \sqrt {{7^2} \times {3^2}} \]
Again, apply the property – squares and square root cancels each other.
 $
  \therefore m = 7 \times 3 \\
  \therefore m = 21 \\
  $
Therefore, the required solution is – If $ {(35)^2} - {\left( {28} \right)^2} = {m^2} $ . Then the value of $ m = 21 $
So, the correct answer is “Option 2”.

Note: The square root of the number “n” is the number when multiplied by the number itself and equals to “n”. For example, the square root of $ \sqrt 9 = \sqrt {{3^2}} = 3 $ . The squares and the square roots are opposite to each other and so cancel each other. Perfect square number is the square of an integer, simply it is the product of the same integer with itself. For example - $ {\text{16 = 4 }} \times {\text{ 4, 16 = }}{{\text{4}}^2} $ , generally it is denoted by n to the power two i.e. $ {n^2} $ . The perfect square is the number which can be expressed as the product of the two equal integers. For example: $ 9 $ , it can be expressed as the product of equal integers. $ 9 = 3 \times 3 $