Mathematical exponent Power rule

Power rule I

(Aⁿ) ^m = A ^{(n x m)}

For example,

(2³)² = 2^3x2 = 2⁶ = 2x2x2x2x2x2 = 64

Power rule II

A^{n ^ m} = A ^{(n ^ m)}

For example,

2^{3^2} = 2^{(3^2)} = 2^(3x3) = 2⁹ = 2x2x2x2x2x2x2x2x2 = 512

Power rule with radicals

^m√ (Aⁿ) = A ^n/m

For example,

²√ (2⁶) = 2^6/2 = 2³ = 2x2x2 = 8

Mathematical exponent Quotient rule

Quotient rule with same base

Aⁿ / A^m = A^n-m

For example,

2⁴ / 2² = 2^4-2 = 2² = 2x2 = 4

           

Quotient rule with same exponent

Aⁿ / Zⁿ = (A / Z) ⁿ

For example,

9² / 3² = (9/3)² = 3² = 3x3 = 9

Mathematical exponent Product rule

Product rule with same base

Aⁿ x A^m = A ^{n + m}

For example,

2² x 2³ = 2²⁺³ = 2⁵ = 2x2x2x2x2 = 32

Product rule with same exponent

Aⁿ x Zⁿ = (A x Z) ⁿ

For example,

2² x 5² = (2x5)² = 10² = 10x10 = 100

Basic concept of mathematical exponent

Basic concept of mathematical exponent represents the base A is raised to the power of n is equal to the mathematical multiplication of the base A up to n times

Aⁿ = A × A × ... × A … up to n times

Where, A is the base and n is the exponent.

For example,

2¹ = 2

2² = 2 × 2 = 4

2³ = 2 × 2 × 2 = 8

2⁴ = 2 × 2 × 2 × 2 = 16 

2⁵ = 2 × 2 × 2 × 2 × 2 = 32

Short cut cubing of any three digit number just in few seconds with help of Horner`s method for polynomial evaluation

We can utilize following algebraic expansion of Horner`s method for polynomial evaluation to easily find cube of any three digit number

(z + d) ³ = z x [z x (z + 3d) + 3d²] + d³

Where, d will always be one of the numbers between (+/-) 1 to (+/-) 50.

For example,

To find cube of the number 323,

Let, z = 300 and d = 23

So,

(z + d) ³ = z x [z x (z + 3d) + 3d²] + d³

(300 + (23)) ³ = 300 x [300 x (300 + 3(23)) + 3(23)²] + (23)³

(323)³ = 300 x [300 x (300 + 69) + 3(23)²] + (23)³

= 300 x [300 x 369 + 3(23)²] + (23)³

= 300 x [300 x 369 + 1587] + (23)³

= 300 x [110700 + 1587] + (23)³

= 300 x 112287 + (23)³

= 33686100 + 12167

= 33698267