1. The power of a real number \(a\) with a natural exponent \(n\) is defined as \({a^n} = \underbrace {a \cdot a \ldots a}_{n\text{ times}},\) where \(a \in \mathbb{R},\) \(n \in \mathbb{N}\).

Properties of powers with natural exponents

1. Multiplication of powers with the same base
   \({a^n}{a^m} = {a^{n + m}}\)
2. Division of powers with the same base
   \({a^n}/{a^m} = {a^{n – m}}\) \(\left( {n \gt m} \right)\)
3. Power of a product
   \({\left( {ab} \right)^n} = {a^n}{b^n}\)
4. Power of a quotient
   \({\left( {\large\frac{a}{b}}\normalsize \right)^n} = {\large\frac{{{a^n}}}{{{b^n}}}\normalsize}\;\) \(\left( {b \ne 0} \right)\)
5. Raising a power to another power
   \({\left( {{a^n}} \right)^m} = {a^{nm}}\)
6. \({0^n} = 0\)
7. \({1^n} = 1\)
8. \({a^1} = a\)
9. Raising a negative number to an even power
   \({\left( { – a} \right)^{2n}} = {a^{2n}}\;\) \(\left( {a \gt 0} \right)\)
10. Raising a negative number to an odd power
    \({\left( { – a} \right)^{2n + 1}} = – {a^{2n + 1}}\;\) \(\left( {a \gt 0} \right)\)

Properties of powers with integer exponents

1. Zero power
   \({a^0} = 1\;\;\left( {a \ne 0} \right)\)
2. The expression \({0^0}\) is not defined.
3. Negative power
   \({a^{ – r}} = 1/{a^r},\) where \(r \in \mathbb{Q},\) \(a \ne 0\).

Properties of powers with rational exponents

1. The power of a positive real number \(a\) with a rational exponent \(p/q\) is defined as\({a^{p/q}} = \sqrt[\large q\normalsize]{{{a^p}}},\) where \(a \ge 0,\) \(p \in \mathbb{Z},\) \(q \in \mathbb{N}\).

Properties of powers with real exponents

1. Definition of power with an irrational exponent \(\beta:\)
   \({a^\beta } = \lim\limits_{{r_n} \to \beta } {a^{{r_n}}},\)
   where \({r_n}\) is an arbitrary sequence of rational numbers converging to the exponent \(\beta\).
2. For any real exponents \(u\), \(v\) under condition of \(a \gt 0\) and \(b \gt 0\) the following operations with powers are valid:
   \({a^u}{a^v} = {a^{u + v}},\) \({\left( {{a^u}} \right)^v} = {a^{uv}},\) \({a^{ – u}} = 1/{a^u},\) \(\large\frac{{{a^u}}}{{{a^v}}}\normalsize = {a^{u – v}},\) \({\left( {ab} \right)^u} = {a^u}{b^u},\) \({\left( {\large\frac{a}{b}}\normalsize \right)^u} = \large\frac{{{a^u}}}{{{b^u}}}\normalsize .\)