# Operations with Powers

• 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): $$n$$, $$m$$, $$r$$, $${r_n}$$, $$\beta$$, $$u$$, $$v$$
Natural numbers: $$n$$, $$m$$, $$q$$
Integers: $$p$$
Rational numbers: $$r$$, $${r_n}$$
Irrational numbers: $$\beta$$
Real numbers: $$u$$, $$v$$
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 $$a \in \mathbb{R},$$ $$n \in \mathbb{N}$$.
2. ### Properties of powers with natural exponents

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

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

### Properties of powers with rational exponents

17. 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 $$a \ge 0,$$ $$p \in \mathbb{Z},$$ $$q \in \mathbb{N}$$.
18. ### Properties of powers with real exponents

19. 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 $$\beta$$.
20. For any real exponents $$u$$, $$v$$ under condition of $$a \gt 0$$ and $$b \gt 0$$ the following operations with powers are valid:
$${a^u}{a^v} = {a^{u + v}},$$ $${\left( {{a^u}} \right)^v} = {a^{uv}},$$ $${a^{ – u}} = 1/{a^u},$$ $$\large\frac{{{a^u}}}{{{a^v}}}\normalsize = {a^{u – v}},$$ $${\left( {ab} \right)^u} = {a^u}{b^u},$$ $${\left( {\large\frac{a}{b}}\normalsize \right)^u} = \large\frac{{{a^u}}}{{{b^u}}}\normalsize .$$