**1. ****Squares:**

1.1. Squares ending in 5:

Separate >> a | 5

Answer : a(a+1) | 25

Eg 35^{2 }>> 3 | 5^{ }>> (3 X 4) | 25 >>1225

Sly, 45^{2} >> 4 | 5 >> 20|25 >> 2025

1.2. General case:

Use Cross-multiply method of multiplication explained in previous post.

**2. ****Cubes: (2 digit no >> ab ^{3})**

{a + b)^{3} = a^{3} + 3a^{2}b + 3ab^{2} + b^{3}

= a^{3} + a^{3}(b/a)^{1} + a^{3}(b/a)^{2} + a^{3}(b/a)^{3} ……… (1)

+ 2 a^{2}b + 2 ab^{2} …….. (2)

__Modus Operandi:__

i. Split no as a | b

ii. Find a^{3} and ratio b/a

iii. From this we get eq (1) by

We get next no by multiplying current no by the ratio b/a.

Left most no is a^{3}.

iv. We get eq (2) by multiplying middle two terms of eq (1) by 2

v. Add only corresponding terms.

Eg. 27^{3}

Here a = 2, b = 3

Therefore, a^{3} = 8 , b/a = 7/2 (=3.5)

Therefore eq (1) = 8 | 28 | 98 | 343

(8 X 3.5 = 28, 28 X 3.5 = 98, 98 X 3.5 = 343)

eq (2) = | 56 | 196 |

(28 X 2 = 56, 98 X 2 = 196)

Sum = 8 | 84 | 294 | 343

Since only 1 digit can be inside on bar, carry forward

8 | _{8}4 | _{29}4 | _{34}3

8 | _{8}4 | _{32}8 | 3

8 | _{11}6 | 8 | 3

19 | 6 | 8 | 3

Therefore, 27^{3} = 19683

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