**Math Exponentiation Formulas**

What is **Exponentiation** ?

**Answer** : Exponentiation is a mathematical operation . It is used as square and multiply algorithm, which is written as x^{a}, it is involving two numbers, the base x and the exponent a. When a is a positive integer, exponentiation corresponds to repeatedly multiplication of the base that is, x^{a} is the product of multiplying a bases:

**Multiplication :** x^{a} . x^{b} = x ^{a + b } Add exponent

**Example :** 3^{3} * 3^{4} = 3^{3+4} = 3^{7}

**Division :** x^{a} / x^{b} = x^{a – b } Subtract exponent

**Example :** x^{7} / x^{5} = x^{7-5} =x^{2}

**Power of power :** (x^{a})^{b} = x^{ab} Multiply exponent

**Example :** (5^{2})^{3} = 5^{2*3} = 5^{6}

**Power of Product :** (xy)^{a} = x^{a} y ^{a} Multiply exponent

**Power of fraction : **(x/y)^{a} = x^{a} / y^{a}

**Inverse :**x^{-a}= 1 / x^{a}

**Zero power :**x^{0}= 1 (x != 0)

- Why a number raised to the 0 power is ?

x / x = 1

x / x = x^{1}/ x^{1}= 1

x^{1}/ x^{1}= x^{1-1}= x^{0}= 1

- X
^{1/n}= n√a

- x
^{m/n}= n√a^{m}

**If the exponent is power of 2**

**Example 1:** Find 3^{128} mod 7

3^{2} mod 7 = 9 mod 7 = 2

3^{4} mod 7 = 9^{2} mod 7 = 81 mod 7 = 4

3^{8} mod 7 = 4^{2} mod 7 = 16 mod 7 = 2

3^{16} mod 7 = 2^{2} mod 7 = 4

3^{32} mod 7 = 4^{2} mod 7 = 2

3^{64} mod 7 = 2^{2} mod 7 = 4

3^{128} mod 7 = 4^{2} mod 7 = 2

So, 3^{128} mod 7 = 2.

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