Question

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

Answer 1

(i) Let us consider the following expression:

$\frac{{58}^2-{42}^2}{16}$
Using the identity $\left(a+b\right)\left(a-b\right)=a^2-b^2$, we get:

$\frac{{58}^2-{42}^2}{16}=\frac{\left(58+42\right)\left(58-42\right)}{16}$
$$\begin{gathered} \Rightarrow \frac{{58}^2-{42}^2}{16}=\frac{100\times 16}{16} \\ \Rightarrow \frac{{58}^2-{42}^2}{16}=100 \end{gathered}$$

Thus, the answer is 100.

(ii) Let us consider the following expression:

$178\times 178-22\times 22$
Using the identity $\left(a+b\right)\left(a-b\right)=a^2-b^2$, we get:

$178\times 178-22\times 22={178}^2-{22}^2=\left(178+22\right)\left(178-22\right)=200\times 156=31200$
Thus, the answer is 31200.

(iii) Let us consider the following expression:

$\frac{198\times 198-102\times 102}{96}=\frac{{198}^2-{102}^2}{96}$
Using the identity $\left(a+b\right)\left(a-b\right)=a^2-b^2$, we get:

$\frac{198\times 198-102\times 102}{96}=\frac{{198}^2-{102}^2}{96}=\frac{\left(198+102\right)\left(198-102\right)}{96}$
$$\begin{gathered} \Rightarrow \frac{198\times 198-102\times 102}{96}=\frac{\left(198+102\right)\left(198-102\right)}{96} \\ \Rightarrow \frac{198\times 198-102\times 102}{96}=\frac{300\times 96}{96} \\ \Rightarrow \frac{198\times 198-102\times 102}{96}=300 \end{gathered}$$

Thus, the answer is 300.

(iv) Let us consider the following expression:

$1.73\times 1.73-0.27\times 0.27$
Using the identity $\left(a+b\right)\left(a-b\right)=a^2-b^2$, we get:

$1.73\times 1.73-0.27\times 0.27=1.{73}^2-0.{27}^2=\left(1.73+0.27\right)\left(1.73-0.27\right)=2\times 1.46=2.92$
Thus, the answer is 2.92.

(v) Let us consider the following expression:

$\frac{8.63\times 8.63-1.37\times 1.37}{0.726}=\frac{8.{63}^2-1.{37}^2}{0.726}$
Using the identity $\left(a+b\right)\left(a-b\right)=a^2-b^2$, we get:

$\frac{8.63\times 8.63-1.37\times 1.37}{0.726}=\frac{8.{63}^2-1.{37}^2}{0.726}=\frac{\left(8.63+1.37\right)\left(8.63-1.37\right)}{0.726}$
$$\begin{gathered} \Rightarrow \frac{8.63\times 8.63-1.37\times 1.37}{0.726}=\frac{\left(8.63+1.37\right)\left(8.63-1.37\right)}{0.726} \\ \Rightarrow \frac{8.63\times 8.63-1.37\times 1.37}{0.726}=\frac{\left(8.63+1.37\right)\left(8.63-1.37\right)}{0.726} \\ \Rightarrow \frac{8.63\times 8.63-1.37\times 1.37}{0.726}=\frac{10\times 7.26}{0.726} \\ \Rightarrow \frac{8.63\times 8.63-1.37\times 1.37}{0.726}=\frac{10\times {\cancel{7.26}}^{10}}{\cancel{0.726}} \\ \Rightarrow \frac{8.63\times 8.63-1.37\times 1.37}{0.726}=100 \end{gathered}$$

Thus, the answer is 100.