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?

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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`.

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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`.

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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`.

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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))`.

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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.

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