Question 1

We will use algebraic identities to simplify these square calculations.

Step 1 — Calculate $117^2$

Let's express 117 as a sum. We can write $117 = 110 + 7$. We will use the identity $(a + b)^2 = a^2 + 2ab + b^2$. Here, $a = 110$ and $b = 7$.

$117^2 = (110 + 7)^2$

$= 110^2 + 2(110)(7) + 7^2$

$= 12100 + 1540 + 49$

$= 13689$

$\boxed{13689}$

Step 2 — Calculate $78^2$

Let's express 78 as a difference. We can write $78 = 80 - 2$. We will use the identity $(a - b)^2 = a^2 - 2ab + b^2$. Here, $a = 80$ and $b = 2$.

$78^2 = (80 - 2)^2$

$= 80^2 - 2(80)(2) + 2^2$

$= 6400 - 320 + 4$

$= 6084$

$\boxed{6084}$

Step 3 — Calculate $198^2$

Let's express 198 as a difference. We can write $198 = 200 - 2$. We will use the identity $(a - b)^2 = a^2 - 2ab + b^2$. Here, $a = 200$ and $b = 2$.

$198^2 = (200 - 2)^2$

$= 200^2 - 2(200)(2) + 2^2$

$= 40000 - 800 + 4$

$= 39204$

$\boxed{39204}$

Step 4 — Calculate $214^2$

Let's express 214 as a sum. We can write $214 = 200 + 14$. We will use the identity $(a + b)^2 = a^2 + 2ab + b^2$. Here, $a = 200$ and $b = 14$.

$214^2 = (200 + 14)^2$

$= 200^2 + 2(200)(14) + 14^2$

$= 40000 + 5600 + 196$

$= 45796$

$\boxed{45796}$

Step 5 — Calculate $1104^2$

Let's express 1104 as a sum. We can write $1104 = 1100 + 4$. We will use the identity $(a + b)^2 = a^2 + 2ab + b^2$. Here, $a = 1100$ and $b = 4$.

$1104^2 = (1100 + 4)^2$

$= 1100^2 + 2(1100)(4) + 4^2$

$= 1210000 + 8800 + 16$

$= 1218816$

$\boxed{1218816}$

Step 6 — Calculate $1120^2$

Let's express 1120 as a sum. We can write $1120 = 1100 + 20$. We will use the identity $(a + b)^2 = a^2 + 2ab + b^2$. Here, $a = 1100$ and $b = 20$.

$1120^2 = (1100 + 20)^2$

$= 1100^2 + 2(1100)(20) + 20^2$

$= 1210000 + 44000 + 400$

$= 1254400$

$\boxed{1254400}$

Answer

(i) $117^2 = 13689$ (ii) $78^2 = 6084$ (iii) $198^2 = 39204$ (iv) $214^2 = 45796$ (v) $1104^2 = 1218816$ (vi) $1120^2 = 1254400$