Set of natural numbers: \(\mathbb{N}\)

Set of integers: \(\mathbb{Z}\)

Set of rational numbers: \(\mathbb{Q}\)

Set of real numbers: \(\mathbb{R}\)

Bases: \(a\), \(b\)

Powers (exponents): \(n\), \(m\), \(r\), \({r_n}\), \(\beta\), \(u\), \(v\)

Set of integers: \(\mathbb{Z}\)

Set of rational numbers: \(\mathbb{Q}\)

Set of real numbers: \(\mathbb{R}\)

Bases: \(a\), \(b\)

Powers (exponents): \(n\), \(m\), \(r\), \({r_n}\), \(\beta\), \(u\), \(v\)

Natural numbers: \(n\), \(m\), \(q\)

Integers: \(p\)

Rational numbers: \(r\), \({r_n}\)

Irrational numbers: \(\beta\)

Real numbers: \(u\), \(v\)

Integers: \(p\)

Rational numbers: \(r\), \({r_n}\)

Irrational numbers: \(\beta\)

Real numbers: \(u\), \(v\)

- 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 \(a \in \mathbb{R},\) \(n \in \mathbb{N}\).
- Multiplication of powers with the same base

\({a^n}{a^m} = {a^{n + m}}\) - Division of powers with the same base

\({a^n}/{a^m} = {a^{n – m}}\) \(\left( {n \gt m} \right)\) - Power of a product

\({\left( {ab} \right)^n} = {a^n}{b^n}\) - Power of a quotient

\({\left( {\large\frac{a}{b}}\normalsize \right)^n} = {\large\frac{{{a^n}}}{{{b^n}}}\normalsize}\;\) \(\left( {b \ne 0} \right)\) - Raising a power to another power

\({\left( {{a^n}} \right)^m} = {a^{nm}}\) - \({0^n} = 0\)
- \({1^n} = 1\)
- \({a^1} = a\)
- Raising a negative number to an even power

\({\left( { – a} \right)^{2n}} = {a^{2n}}\;\) \(\left( {a \gt 0} \right)\) - Raising a negative number to an odd power

\({\left( { – a} \right)^{2n + 1}} = – {a^{2n + 1}}\;\) \(\left( {a \gt 0} \right)\) - Zero power

\({a^0} = 1\;\;\left( {a \ne 0} \right)\) - The expression \({0^0}\) is not defined.
- Negative power

\({a^{ – r}} = 1/{a^r},\) where \(r \in \mathbb{Q},\) \(a \ne 0\).### Properties of powers with rational exponents

- 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 \(a \ge 0,\) \(p \in \mathbb{Z},\) \(q \in \mathbb{N}\). - 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 \(\beta\). - For any real exponents \(u\), \(v\) under condition of \(a \gt 0\) and \(b \gt 0\) the following operations with powers are valid:

\({a^u}{a^v} = {a^{u + v}},\) \({\left( {{a^u}} \right)^v} = {a^{uv}},\) \({a^{ – u}} = 1/{a^u},\) \(\large\frac{{{a^u}}}{{{a^v}}}\normalsize = {a^{u – v}},\) \({\left( {ab} \right)^u} = {a^u}{b^u},\) \({\left( {\large\frac{a}{b}}\normalsize \right)^u} = \large\frac{{{a^u}}}{{{b^u}}}\normalsize .\)