The properties of the powers of x (for natural exponents)

Product/quotient of powers (having the same base)

1. Product

   Let \( x\in \hspace{0.04em} \mathbb{R} \) a real number and \( (a, b) \in \hspace{0.04em} \mathbb{N}^2 \) two natural numbers.

   If we perform the product \(x^a x^b \), we do have:

   $$ x^a x^b = \hspace{0.2em} \underbrace {x \times x \times x \times x \times x ...} _\text{ \(a\) factors } \ \times \ \hspace{0.2em} \underbrace {x \times x \times x \times x \times x ...} _\text{ \(b\) factors }$$
   $$ x^a x^b = \hspace{0.2em} \underbrace {x \times x \times x \times x \times x ...} _\text{ \((a + b)\) factors } $$

   Considering the fundamental definition of a power of x :

   $$ x^n = \hspace{0.2em} \underbrace {x \times x \times x \times x \times x ...} _\text{ \(n\) factors } $$

   We finally obtain that,

   $$ \forall x \in\hspace{0.04em} \mathbb{R}, \enspace \forall ( a, b) \in \hspace{0.04em} \mathbb{N}^2, $$
   $$ x^a x^b = x^{a+b} $$

2. Quotient

   Since division by a number is only multiplication by the inverse of that same number, the previous product formula also applies to quotients, with the additional condition that \((x \neq 0) \):

   $$ \frac{x^a}{x^b} = x^a \times \frac{1}{x^b} $$
   $$ \frac{x^a}{x^b} = x^a \times x^{-b} $$
   $$ \frac{x^a}{x^b} = x^{a+(-b)} $$
   $$ \forall x \in \hspace{0.03em} \mathbb{R}^*, \ \forall (a, b) \in \hspace{0.03em} \mathbb{N}^2, $$
   $$ \frac{x^a}{x^b} = x^{a-b} $$

Number to the power of zero

Let \( x \in \hspace{0.04em} \mathbb{R}^* \) be a non-zero real number and \( a \in \hspace{0.04em} \mathbb{N} \) a natural number.

With the formula for multiplying powers , we can do this:

By dividing each member by \(x^a\), adding the condition that \((x \neq 0)\) :

And finally,

Inverse of a power

Let \( x \in \hspace{0.04em} \mathbb{R}^* \) be a non-zero real number and \( a \in \hspace{0.04em} \mathbb{N} \) a natural number.

Let's find out what the power of the inverse would be, we know that:

But \(x^0 = 1\), so :

Therefore, with the formula for multiplying powers , we dohave now:

So by identifying the formulas \((3)\) and \((4)\), we obtain as a result that:

Power of a product/quotient

1. Product

   Let \( (x, y) \in \hspace{0.04em} \mathbb{R}^2 \) be two real numbers and \( a\in \hspace{0.04em} \mathbb{N} \) a natural number.

   $$ (xy)^a = \hspace{0.2em} \underbrace {xy \times xy \times xy \times xy \times xy ...} _\text{ \(a\) factors } $$

   The product of real numbers being commutative, we have:

   $$ (xy)^a = \hspace{0.2em} \underbrace {x \times x \times x\times x \times x...} _\text{ \(a\) factors } \ \times \ \underbrace {y \times y \times y \times y \times y ...} _\text{ \(a\) factors } $$

   Considering the fundamental definition of a power of x:

   $$ x^n = \hspace{0.2em} \underbrace {x \times x \times x \times x \times x ...} _\text{ \(n\) factors } $$

   We now have,

   $$ (xy)^a = x^a \times y^a $$

   So finally,

   $$ \forall (x,y) \in \hspace{0.04em} \mathbb{R}^2, \ \forall a \in \mathbb{N}, $$
   $$ (xy)^a = x^a y^a $$

2. Quotient

   Let \( x \in \mathbb{R} \) be a real number, \( y \in \hspace{0.03em} \mathbb{R}^* \) a non-zero real number and \( a\in \hspace{0.04em} \mathbb{N} \) a natural number.

   Since division by a number is only multiplication by the inverse of that same number, the previous product formula also applies to quotients, with the additional condition that \((x \neq 0) \):

   $$ \frac{x^a}{y^a} = x^a \times \left(\frac{1}{y}\right)^a $$

   So, the same goes for quotients,

   $$ \forall x \in \mathbb{R}, \ \forall y \in \mathbb{R}^*, \ \forall a \in \hspace{0.04em} \mathbb{N} , $$
   $$ \left(\frac{x}{y}\right)^a = \frac{x^a}{y^a} $$

Power of a power

Soient \( x\in \hspace{0.04em} \mathbb{R} \) a real number and \( (a, b) \in \hspace{0.04em} \mathbb{N}^2 \) two natural numbers.

If we perform the computation \( (x^a)^b \), we obtain that:

So we find that,

y applying this same reasoning but reversing \(a\) and \(b\), we also find that:

Thus,

Recap table of the formulas of the powers of x