Find values using suitable identities:

(A) $17 \times 21$

(B) $104 \times 96$

(C) $24 \times 16$

(D) $147^3$

(E) $199^3$

(F) $127^3$

(G) $(-107)^3$

(H) $(-299)^3$

(A) $17 \times 21 = (19 - 2)(19 + 2)$

$$= 19^2 - 2^2$$

$$= 361 - 4 = 357$$

(B) $104 \times 96 = (100 + 4)(100 - 4)$

$$= 10000 - 16 = 9984$$

Concept Applied
The difference of two squares $a^2 - b^2$ factorises as $(a + b)(a - b)$ .

(C) $24 \times 16 = (20 + 4)(20 - 4)$

$$= 400 - 16 = 384$$

(D) $147^3 = (150 - 3)^3$

$$= 150^3 - 3 \times 150^2 \times 3 + 3 \times 150 \times 9 - 27$$

$$= 3375000 - 202500 + 4050 - 27$$

$$= 3176523$$

(E) $199^3 = (200 - 1)^3$

$$= 8000000 - 3(40000) + 3(200) - 1$$

$$= 8000000 - 120000 + 600 - 1$$

$$= 7880599$$

(F) $127^3 = (130 - 3)^3$

$$= 130^3 - 3 \times 130^2 \times 3 + 3 \times 130 \times 9 - 27$$

$$= 2197000 - 152100 + 3510 - 27$$

$$= 2048383$$

(G) $(-107)^3 = -(107)^3 = -(100 + 7)^3$

$$(107)^3 = (100 + 7)^3$$

$$= 1000000 + 3(10000)(7) + 3(100)(49) + 343$$

$$= 1000000 + 210000 + 14700 + 343$$

$$= 1225043$$

$$\text{So, } (-107)^3 = -1225043$$

(H) $(-299)^3 = -(299)^3 = -(300 - 1)^3$

$$299^3 = 27000000 - 3(90000) + 3(300) - 1$$

$$= 27000000 - 270000 + 900 - 1$$

$$= 26730899$$

$$\text{So, } (-299)^3 = -26730899$$