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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, $9.8 \times 10.2=99.96$.
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