Evaluate without multiplying directly 998³ x 103³​

1,086,183,741,982,184

Step-by-step explanation:

998^3

 = (1,000)^3 - (2)^3 - 3 · 1,000 · 2 (1000 - 2)

 = 1,000,000,000 - 8 - 6,000 (998)

 = 999,999,992 - 5,988,000

 = 994,011,992

103^3

 = (100 + 3)^3

 = (100)^3 + (3)^3 + 3(100) (3) + (100+3)

 = 1,000,000 + 27 + 900(103)

 = 1,000,027 + 92,700

 = 1,092,727

994,011,992 · 1,092,727

 = 994,011,992 · (1,000,000 + 90,000 + 2,000 + 700 + 20 + 7)

 = (994,011,992 · 1,000,000) + (994,011,992 · 90,000) + (994,011,992 · 2,000) + (994,011,992 · 700) + (994,011,992 · 20) + (994,011,992 · 7)

= 994,011,992,000,000 + 89,461,079,280,000 + 1,988,023,984,000 + 695,808,394,400 + 19,880,239,840 + 6,958,083,944

 = 1,086,183,741,982,184

998 can be written as (1000 – 2)and103^3 can be written as 100+3

∴ 998 = 1000 – 2∴103=100+3

(998)³ = (1000 – 2)³

(103)³=(100+3)³

We know the formulas

(a – b)³ = a³ – b³ – 3ab(a – b)

(a + b)³ = a³ + b³ + 3ab(a + b)

(1000 – 2)³×(100+3)³ = {(1000)³ – (2)³ – 3 × 1000 × 2(1000 – 2)}×{(100)³+(3)³+3(100+3)}

(998)³×(103)³ =( 1000000000 − 8 – 5988000)×(1000000+27+309)

(998)³×(103)³ = 994011992×1000336

(998)³×(103)³=994345980000000

(998)³×(103)³=9.9434598×10^14