# Category: Powers and Exponents

## Fractional (Rational) Exponents

Fractional exponent is a natural extension to the integer exponent.

We already know, that if b is positive integer, then

• $$a^b=\underbrace{a \cdot a \cdot a \cdot...\cdot a}_{b}$$$(see positive exponent) • $$a^{-b}=\frac{1}{a^b}=\frac{1}{\underbrace{a \cdot a \cdot a \cdot...\cdot a}_{b}}$$$ (see negative exponets)

But what if exponent is a fraction?

To understand addition of exponents, let's start from a simple example.

Example. Suppose, we want to find 2^3*2^4.

We already learned about positive integer exponets, so we can write, that 2^3=2*2*2 and 2^4=2*2*2*2.

## Subtracting Exponents

To understand subtraction of exponents, let's start from a simple example.

Example. Suppose, we want to find (2^7)/(2^4).

We already learned about positive integer exponets, so we can write, that 2^7=2*2*2*2*2*2*2 and 2^4=2*2*2*2.

## Multiplying Exponents

To understand multiplication of exponents, let's start from a simple example.

Example. Suppose, we want to find (2^3)^4.

We already learned about positive integer exponets, so we can rewrite outer exponent: (2^3)^4=2^3*2^3*2^3*2^3.

## Dividing Exponents

We already saw division of exponents two times:

• when discussed fractional exponents (a^(m/n)=root(n)(a^m))
• when discussed multiplication of exponents (indeed, a^(m/n)=a^(m*1/n)=(a^m)^(1/n)=root(n)(a^m)).

Rule for dividing exponents: huge color(purple)(root(n)(a^m)=a^(m/n)).

## Properties of Exponents (Rules)

Properties (rules) of exponents:

• Zero power: a^0=1, a!=0
• Zero base: 0^a=0, a!=0
• 0^0 is undefined
• 1^a=1
• Negative exponent: a^(-b)=1/a^b, b!=0
• Nth root: a^(1/n)=root(n)(a), n!=0
• Addition of exponents: a^m*a^n=a^(m+n)
• Subtraction of exponents: (a^m)/(a^n)=a^(m-n), a!=0
• Multiplication of exponents: (a^m)^n=a^(m*n)=(a^n)^m
• Division of exponents: root(n)(a^m)=a^(m/n), n!=0
• root(m)(a^m)=a, if m is odd
• root(m)(a^m)=|a|, if m is even
• root(n)(a^m)=(root(n)(a))^m (just pay attention to signs and check, whether number exists)
• Power of a product: a^n*b^n=(ab)^n
• Power of a quotient: (a^n)/(b^n)=(a/b)^n, b!=0

We already covered all rules earlier, except last two.