Question

Using ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$ , find
(i) ${51^2} - {49^2}$
(ii) ${1.02^2} - {0.98^2}$
(iii) ${153^2} - {147^2}$
(iv) ${12.1^2} - {7.9^2}$

Answer 1

Hint: In this question we just need to use the equation given in the question. We will find the values of $a$ and $b$ from the question given to us and then we will put those values into the equation given in the question and then we have to just simplify the equation.

Complete step-by-step solution:
(i) ${51^2} - {49^2}$
We have apply the equation ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$ to $
  {51^2} - {49^2} \\
    \\
 $. Here, we can compare ${a^2} - {b^2}$ with $
  {51^2} - {49^2} \\
    \\
 $. Therefore, by the comparison we get $a = 51$ and $b = 49$
Now, put $a = 51$ and $b = 49$ in the equation ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$. Therefore, we can write ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$ as follows.
$
  {51^2} - {49^2} = \left( {51 + 49} \right)\left( {51 - 49} \right) \\
   = \left( {100} \right)\left( 2 \right) = 200
 $
Therefore, ${51^2} - {49^2} $ is $200$.
(ii) ${1.02^2} - {0.98^2}$
We have apply the equation ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$ to ${1.02^2} - {0.98^2}$. Here, we can compare ${a^2} - {b^2}$ with ${1.02^2} - {0.98^2}$. Therefore, by the comparison we get $a = 1.02$ and $b = 0.98$
Now, put $a = 1.02$ and $b = 0.98$ in the equation ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$. Therefore, we can write ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$ as follows.
$
  {1.02^2} - {0.98^2} = \left( {1.02 + 0.98} \right)\left( {1.02 - 0.98} \right) \\
   = \left( 2 \right)\left( {0.04} \right) = 0.08
 $
Therefore, ${1.02^2} - {0.98^2}$ is $0.08$.
(iii) ${153^2} - {147^2}$
We have apply the equation ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$ to ${153^2} - {147^2}$. Here, we can compare ${a^2} - {b^2}$ with ${153^2} - {147^2}$. Therefore, by the comparison we get $a = 153$ and $b = 147$
Now, put $a = 153$ and $b = 147$ in the equation ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$. Therefore, we can write ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$ as follows.
$
  {153^2} - {147^2} = \left( {153 + 147} \right)\left( {153 - 147} \right) \\
   = \left( {300} \right)\left( 6 \right) = 1800
 $
Therefore, ${153^2} - {147^2}$ is $1800$.
(iv) ${12.1^2} - {7.9^2}$
We have apply the equation ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$ to ${12.1^2} - {7.9^2}$. Here, we can compare ${a^2} - {b^2}$ with ${12.1^2} - {7.9^2}$. Therefore, by the comparison we get $a = 12.1$ and $b = 7.9$
Now, put $a = 12.1$ and $b = 7.9$ in the equation ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$. Therefore, we can write ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$ as follows.
$
  {12.1^2} - {7.9^2} = \left( {12.1 + 7.9} \right)\left( {12.1 - 7.9} \right) \\
   = \left( {20} \right)\left( {4.2} \right) = 84
 $
Therefore, ${12.1^2} - {7.9^2}$ is $84$.

Note: In this question we just need to be careful when choosing the value for $a$ and $b$. Because if we have chosen the wrong value for $a$ and $b$ then our final answer will be incorrect. And also be careful while putting the value with a sign. If the value is negative, put that value with a sign.