Question

Simplify the following using identities.
a.${\left( {109} \right)^2} + {\left( {91} \right)^2}$
b.${\left( {200} \right)^2} - {\left( {100} \right)^2}$
c.${\left( {1025} \right)^2} - {\left( {975} \right)^2}$
d.${\left( {10} \right)^2} + {\left( {20} \right)^2}$

Answer 1

Firstly, observe the question and then use the identity ${\left( {a + b} \right)^2} + {\left( {a - b} \right)^2} = 2\left[ {{a^2} + {b^2}} \right]$ , ${a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)$ , ${\left( {a + b} \right)^2} - {\left( {a - b} \right)^2} = 4ab$ , ${a^2} + {b^2} = {\left( {a + b} \right)^2} - 2ab$ . With the formula we can easily simplify this equation by knowing all the variables i.e a and b.

Formula Used: Here, we can use the formula according to the requirement of the question${\left( {a + b} \right)^2} + {\left( {a - b} \right)^2} = 2\left[ {{a^2} + {b^2}} \right]$ , ${a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)$ , ${\left( {a + b} \right)^2} - {\left( {a - b} \right)^2} = 4ab$ , ${a^2} + {b^2} = {\left( {a + b} \right)^2} - 2ab$

Complete step-by-step answer:
Now, we will begin with:
(a)${\left( {109} \right)^2} + {\left( {91} \right)^2}$
First, split 109 and 91 in factors of 100.
$\Rightarrow {\left( {100 + 9} \right)^2} + {\left( {100 - 9} \right)^2}$
Here, it becomes an identity ${\left( {a + b} \right)^2} + {\left( {a - b} \right)^2} = 2\left[ {{a^2} + {b^2}} \right]$
Now considering left hand side , {\left( {a + b} \right)^2} + {\left( {a - b} \right)^2} = $$$${\left( {100 + 9} \right)^2} + {\left( {100 - 9} \right)^2}
From here, we can clearly see that $a = 100$ and $b = 9$ .
Put the values of a and b in right hand side of formula $\Rightarrow 2\left[ {{a^2} + {b^2}} \right]$
$\Rightarrow 2\left[ {{{\left( {100} \right)}^2} + {{\left( 9 \right)}^2}} \right]$
By opening the squares:
$\Rightarrow 2\left[ {10000 + 81} \right]$
On further simplifying:
$\Rightarrow 2\left[ {10081} \right]$
We get, $\Rightarrow 20162$
{\left( {109} \right)^2} + {\left( {91} \right)^2}$$$$ \Rightarrow 20162

(b)${\left( {200} \right)^2} - {\left( {100} \right)^2}$
As, it is clearly visible that is ${a^2} - {b^2}$ and hence we can use the identity ${a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)$ .
Now considering left hand side , {a^2} - {b^2}$$$$ = {\left( {200} \right)^2} - {\left( {100} \right)^2}
From here, we can clearly see that $a = 200$ and $b = 100$ .
Put the values of a and b in right hand side of formula $\Rightarrow \left( {a - b} \right)\left( {a + b} \right)$
$\Rightarrow \left( {200 - 100} \right)\left( {200 + 100} \right)$
On simplifying:
$\Rightarrow 100 * 300$
We get, $\Rightarrow 30000$
{\left( {200} \right)^2} - {\left( {100} \right)^2}$$$$ \Rightarrow 30000

(c)${\left( {1025} \right)^2} - {\left( {975} \right)^2}$
First, of all split 1025 and 975 in factors of 1000.
$\Rightarrow {\left( {1000 + 25} \right)^2} - {\left( {1000 - 25} \right)^2}$
Here, it becomes an identity ${\left( {a + b} \right)^2} - {\left( {a - b} \right)^2} = 4ab$
Now considering left hand side , {\left( {a + b} \right)^2} - {\left( {a - b} \right)^2} = $$$${\left( {1000 + 25} \right)^2} + {\left( {1000 - 25} \right)^2}
From here, we can clearly see that $a = 1000$ and $b = 25$ .
Put the values of a and b in right hand side of formula $\Rightarrow 4ab$
$\Rightarrow 4*1000*25$
By multiplying:
$\Rightarrow 100000$
{\left( {1025} \right)^2} - {\left( {975} \right)^2}$$$$ \Rightarrow 100000

(d)${\left( {10} \right)^2} + {\left( {20} \right)^2}$
As, it is clearly visible that is ${a^2} + {b^2}$ and hence we can use the identity ${a^2} + {b^2} = {\left( {a + b} \right)^2} - 2ab$
Now considering left hand side , {a^2} + {b^2}$$$$ = {\left( {10} \right)^2} + {\left( {20} \right)^2}
From here, we can clearly see that $a = 10$ and $b = 20$ .
Put the values of a and b in right hand side of formula $= {\left( {a + b} \right)^2} - 2ab$
$\Rightarrow {\left( {10 + 20} \right)^2} - 2 * 10 * 20$
$\Rightarrow {\left( {30} \right)^2} - 2 * 10 * 20$
On simplifying square,
$\Rightarrow 900 - 2 * 10 * 20$
And on multiplying,
$\Rightarrow 900 - 400$
We get, $\Rightarrow 500$
{\left( {10} \right)^2} + {\left( {20} \right)^2}$$$$ \Rightarrow 500

Note: In this type of question, first of all observe the given statements and calculate the values. Accordingly, implement the identities which are most appropriate.