Number Operations and Properties

28 * 22 = 616. Using the Nikhilam formula or standard multiplication, the result is 616.

Dhwajanka method calculations: 3987/28 gives Q=142, R=11; 5786/78 gives Q=74, R=14; 7396/82 gives Q=90, R=16.

46 * 4 = 184. The vinculum method represents 46 as 5(bar 4), and the multiplication follows standard arithmetic.

Deviations from 10 are +2, +3, +7. Product = (12+3+7) | (2*3 + 3*7 + 7*2) | (2*3*7) = 22 | (6+21+14) | 42 = 22 | 41 | 42 = 2652.

Using base 10, deviations are +2 and +8. Product = (12+8) | (2*8) = 20 | 16 = 216.

The cubage (cube) of 71 using the Nikhilam formula involves base 70. 71^3 = (71 + 2*1) * 70^2 + 3*1^2 * 70 + 1^3 = 73 * 4900 + 210 + 1 = 357700 + 211 = 357911.

$={8}^{3+2-5}$

$(10)^{150}\div (10)^{146} = \dfrac{10^{150}}{10^{146}}$
$= 10^{150-146}$
$= 10^4$
$= 10000.$

${ 10 }^{ -49 }-2\cdot { 10 }^{ -50 }={ 10 }^{ -50 }(10-2)=8\cdot { 10 }^{ -50 }\ \therefore { 10 }^{ -49 }\hspace{1mm} exceeds\hspace{1mm} 2\cdot { 10 }^{ -50 }\hspace{1mm} by\hspace{1mm} 8\cdot { 10 }^{ -50 }$

$2^{n-m}=16\2^{n-m}=2^4\n-m=4\cdots(1)\3^{n+m}=729\3^{n+m}=3^6\n+m=6\cdots(2)\(1)+(2)\2n=10\\boxed{n=5}\\boxed{m=1}$
$\boxed{nm=5}...Answer$

$(32)(32)^{1/6}(32)^{1/36}......$ to $\infty = (32)^{1+\dfrac{1}{6}.....\infty}$

Zeroes after decimal but after non zero value are significant.