Simplify the following using the identities:
(i) $\frac{((58)^2 – (42)^2)}{16}$
(ii) $178 \times 178 – 22 \times 22$
(iii) $\frac{(198 \times 198 – 102 \times 102)}{96}$
(iv) $1.73 \times 1.73 – 0.27 \times 0.27$
(v) $\frac{(8.63 \times 8.63 – 1.37 \times 1.37)}{0.726}$

Given:

(i) $\frac{((58)^2 – (42)^2)}{16}$ 

(ii) $178 \times 178 – 22 \times 22$ 

(iii) $\frac{(198 \times 198 – 102 \times 102)}{96}$ 

(iv) $1.73 \times 1.73 – 0.27 \times 0.27$ 

(v) $\frac{(8.63 \times 8.63 – 1.37 \times 1.37)}{0.726}$

To do:

We have to simplify the given expressions using suitable identities.

Solution:

Here, we have to simplify the given expressions. The given expressions(numerators in the expressions) are in the form of difference of two square numbers. We can simplify the given expressions by using the identity $a^2-b^2=(a+b) \times (a-b)$.

(i) The given expression is $\frac{((58)^2 – (42)^2)}{16}$ 

Here, $a=58$ and $b=42$

Therefore,

$\frac{((58)^2 – (42)^2)}{16}=\frac{(58+42) \times (58-42)}{16}$

$\frac{((58)^2 – (42)^2)}{16}=\frac{100\times16}{16}$

$\frac{((58)^2 – (42)^2)}{16}=100$

Hence, $\frac{((58)^2 – (42)^2)}{16}=100$.

(ii) The given expression is $178 \times 178 – 22 \times 22$

$178 \times 178 – 22 \times 22=(178)^2-(22)^2$

Here, $a=58$ and $b=42$

Therefore,

$178 \times 178 – 22 \times 22=(178)^2-(22)^2$

$178 \times 178 – 22 \times 22=(178+22) \times (178-22)$

$178 \times 178 – 22 \times 22=200\times156$

$178 \times 178 – 22 \times 22=31200$

(iii) The given expression is $\frac{(198 \times 198 – 102 \times 102)}{96}$ 

The numerator can be written as $198 \times 198 – 102 \times 102=(198)^2-(102)^2$

Here, $a=198$ and $b=102$

Therefore,

$\frac{((198)^2 – (102)^2)}{96}=\frac{(198+102) \times (198-102)}{96}$

$\frac{((198)^2 – (102)^2)}{96}=\frac{300\times96}{96}$

$\frac{((198)^2 – (102)^2)}{96}=300$

Hence, $\frac{(198 \times 198 – 102 \times 102)}{96}=300$.

(iv) The given expression is $1.73 \times 1.73 – 0.27 \times 0.27$ 

$1.73 \times 1.73 – 0.27 \times 0.27=(1.73)^2-(0.27)^2$

Here, $a=1.73$ and $b=0.27$

Therefore,

$1.73 \times 1.73 – 0.27 \times 0.27=(1.73)^2-(0.27)^2$

$1.73 \times 1.73 – 0.27 \times 0.27=(1.73+0.27) \times (1.73-0.27)$

$1.73 \times 1.73 – 0.27 \times 0.27=2.00\times1.46$

$1.73 \times 1.73 – 0.27 \times 0.27=2.92$

Hence, $1.73 \times 1.73 – 0.27 \times 0.27=2.92$

(v) The given expression is $\frac{(8.63 \times 8.63 – 1.37 \times 1.37)}{0.726}$

The numerator can be written as $8.63 \times 8.63 – 1.37 \times 1.37=(8.63)^2-(1.37)^2$

Here, $a=8.63$ and $b=1.37$

Therefore,

$\frac{((8.63)^2 – (1.37)^2)}{0.726}=\frac{(8.63+1.37) \times (8.63-1.37)}{0.726}$

$\frac{((8.63)^2 – (1.37)^2)}{0.726}=\frac{10.00\times7.26}{0.726}$

$\frac{((8.63)^2 – (1.37)^2)}{0.726}=\frac{72.6}{0.726}$

$\frac{((8.63)^2 – (1.37)^2)}{0.726}=\frac{726\times10^{-1}}{726\times10^{-3}}$

$\frac{((8.63)^2 – (1.37)^2)}{0.726}=10^{-1+3}$

$\frac{((8.63)^2 – (1.37)^2)}{0.726}=10^2$

$\frac{((8.63)^2 – (1.37)^2)}{0.726}=100$

Hence, $\frac{(8.63 \times 8.63 – 1.37 \times 1.37)}{0.726}=100$.