Simplify the following using the formula: $(a - b) (a + b) = a^2 - b^2$:
(i) $(82)^2 – (18)^2$
(ii) $(467)^2 – (33)^2$
(iii) $(79)^2 – (69)^2$
(iv) $197 \times 203$
(v) $113 \times 87$
(vi) $95 \times 105$
(vii) $1.8 \times 2.2$
(viii) $9.8 \times 10.2$

Given:

(i) $(82)^2 – (18)^2$ 

(ii) $(467)^2 – (33)^2$ 

(iii) $(79)^2 – (69)^2$ 

(iv) $197 \times 203$ 

(v) $113 \times 87$ 

(vi) $95 \times 105$ 

(vii) $1.8 \times 2.2$ 

(viii) $9.8 \times 10.2$

To do:

We have to simplify the given expressions using the formula: $(a – b) (a + b) = a^2 – b^2$

Solution:

Here, we have to simplify the given expressions using the formula $(a – b) (a + b) = a^2 – b^2$. The given expressions can be written as the difference of two squares by writing the terms as the sum or difference of two suitable numbers.  

(i) The given expression is $(82)^2 – (18)^2$ 

Here, $a=82$ and $b=18$

Therefore,

$(82)^2 – (18)^2=(82-18)\times(82+18)$

$(82)^2 – (18)^2=64\times100$

$(82)^2 – (18)^2=6400$

Hence, $(82)^2 – (18)^2=6400$.

(ii) The given expression is $(467)^2 – (33)^2$ 

Here, $a=467$ and $b=33$

Therefore,

$(467)^2 – (33)^2=(467-33)\times(467+33)$

$(467)^2 – (33)^2=434\times500$

$(467)^2 – (33)^2=434\times5\times100$                ($500=5\times100$)

$(467)^2 – (33)^2=434\times100\times5$                [$a \times (b \times c)= (a \times c) \times b$]

$(467)^2 – (33)^2=43400\times5$

$(467)^2 – (33)^2=217000$

Hence, $(467)^2 – (33)^2=217000$.

(iii) The given expression is $(79)^2 – (69)^2$ 

Here, $a=82$ and $b=18$

Therefore,

$(79)^2 – (69)^2=(79-69)\times(79+69)$

$(79)^2 – (69)^2=10\times148$

$(79)^2 – (69)^2=1480$

Hence, $(79)^2 – (69)^2=1480$.

(iv) The given expression is $197 \times 203$ 

We can write $197$ as $197=200-3$ and $203$ as $203=200+3$

Here, $a=200$ and $b=3$

Therefore,

$197 \times 203=(200-3)\times(200+3)$

$197 \times 203=(200)^2-(3)^2$

$197 \times 203=40000-9$

$197 \times 203=39991$

Hence, $197 \times 203=39991$.

(v) The given expression is $113 \times 87$ 

We can write $113$ as $113=100+13$ and $87$ as $87=100-13$

Here, $a=100$ and $b=13$

Therefore,

$113 \times 87=(100+13)\times(100-13)$

$113 \times 87=(100)^2-(13)^2$

$113 \times 87=10000-169$

$113 \times 87=9831$

Hence, $113 \times 87=9831$.

(vi) The given expression is $95 \times 105$ 

We can write $95$ as $95=100-5$ and $105$ as $105=100+5$

Here, $a=100$ and $b=5$

Therefore,

$95 \times 105=(100-5)\times(100+5)$

$95 \times 105=(100)^2-(5)^2$

$95 \times 105=10000-25$

$95 \times 105=9975$

Hence, $95 \times 105=9975$.

(vii) The given expression is $1.8 \times 2.2$ 

We can write $1.8$ as $1.8=2-0.2$ and $2.2$ as $2.2=2+0.2$

Here, $a=2$ and $b=0.2$

Therefore,

$1.8 \times 2.2=(2-0.2)\times(2+0.2)$

$1.8 \times 2.2=(2)^2-(0.2)^2$

$1.8 \times 2.2=4-0.04$

$1.8 \times 2.2=3.96$

Hence, $1.8 \times 2.2=3.96$.

(viii) The given expression is $9.8 \times 10.2$

We can write $9.8$ as $9.8=10-0.2$ and $10.2$ as $10.2=10+0.2$

Here, $a=10$ and $b=0.2$

Therefore,

$9.8 \times 10.2=(10-0.2)\times(10+0.2)$

$9.8 \times 10.2=(10)^2-(0.2)^2$

$9.8 \times 10.2=100-0.04$

$9.8 \times 10.2=99.96$

Hence, $9.8 \times 10.2=99.96$.