Exponents (also called powers) are a compact way to write repeated multiplication.

If a is a real number and n is a positive integer, then $a^n$an means a multiplied by itself n times:

$$a^n = \underbrace{a \cdot a \cdot \dots \cdot a}_{n\ \text{factors}}.$$

Here $a$ is the base and $n$ is the exponent (or power).

Basic rules of exponents

1. Product of powers with the same base $$a^m \cdot a^n = a^{m+n}.$$
2. Quotient of powers with the same base $$\frac{a^m}{a^n} = a^{m-n}\quad (a\neq 0).$$
3. Power of a power $$(a^m)^n = a^{m n}.$$
4. Power of a product $$(ab)^n = a^n b^n.$$
5. Power of a quotient $$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\quad (b\neq 0).$$
6. Zero exponent $$a^0 = 1\quad (a\neq 0).$$
7. Negative exponents $$a^{-n} = \frac{1}{a^n}\quad (a\neq 0).$$
8. Fractional (rational) exponents $$a^{\frac{m}{n}} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m\quad (a\ge 0\ \text{if}\ n\ \text{even}).$$
9. Exponent of 1 and -1 $$1^n = 1,\qquad (-1)^n = \begin{cases} 1, & \text{if } n \text{ is even},\\ -1, & \text{if } n \text{ is odd}. \end{cases}$$

Important remarks

• The rules above assume the same base when combining powers (rules 1 and 2). You cannot add exponents if the bases are different (e.g., $a^2 + a^3 \neq a^5$).
• Pay attention to parentheses: $(-2)^3 = -8$ but $-2^3 = -(2^3) = -8$ (actually both equal -8 here), while $(-2)^2 = 4$ and $-2^2 = -4$ (the latter because exponent binds before the unary minus).
• For non-integer exponents of negative bases, results may be complex numbers.

Worked examples

Example 1 — Simplify using exponent rules
Simplify: $x^3 \cdot x^5$

Solution:
Use product rule $x^3 \cdot x^5 = x^{3+5} = x^8.$

Example 2 — Quotient of powers
Simplify: $\dfrac{y^{10}}{y^4}$

Solution:
Use quotient rule $\dfrac{y^{10}}{y^4} = y^{10-4} = y^6.$

Example 3 — Power of a power
Simplify: $(2^3)^4$

Solution:
Use power of a power $(2^3)^4 = 2^{3\cdot 4} = 2^{12} = 4096.$

Example 4 — Power of a product and negative exponent
Simplify: $(3x^2 y)^{-2}$

Solution:
First apply power of a product:

$$(3x^2 y)^{-2} = 3^{-2} (x^2)^{-2} y^{-2} = \frac{1}{3^2} \cdot x^{-4} \cdot y^{-2}.$$

Rewrite negative exponents as reciprocals:

$$= \frac{1}{9} \cdot \frac{1}{x^{4}} \cdot \frac{1}{y^{2}} = \frac{1}{9x^{4}y^{2}}.$$

Example 5 — Fractional exponent
Simplify: $16^{3/4}$

Solution:

Write as root: $16^{3/4} = \left(\sqrt[4]{16}\right)^3$
Since $\sqrt[4]{16} = 2$4 (because $2^4=16$), we get

$$16^{3/4} = 2^3 = 8.$$

Example 6 — Combine several rules
Simplify: $\dfrac{(2a^3 b^{-2})^2}{4 a^{-1} b}$

Solution:
Compute numerator first:

$$(2a^3 b^{-2})^2 = 2^2 \cdot (a^3)^2 \cdot (b^{-2})^2 = 4 \cdot a^{6} \cdot b^{-4}.$$

So the fraction becomes

$$\frac{4 a^{6} b^{-4}}{4 a^{-1} b} = \frac{a^{6} b^{-4}}{a^{-1} b}.$$

Use quotient rule for each base:

$$= a^{6-(-1)} b^{-4-1} = a^{7} b^{-5} = \frac{a^{7}}{b^{5}}.$$

Practice problems (try these)

1. Simplify $(x^4 y^{-1})^3$
2. Evaluate $81^{-1/2}$
3. Simplify $\dfrac{(3x^2)^3}{9x^4}$
4. Simplify $( -2 )^5$and $-2^5$— explain the difference.

Answers

1. $x^{12} y^{-3} = \dfrac{x^{12}}{y^{3}}$
2. $81^{-1/2} = \dfrac{1}{\sqrt{81}} = \dfrac{1}{9}.$
3. Numerator $(3x^2)^3 = 27 x^6$. Then $\dfrac{27 x^6}{9 x^4} = 3 x^{2}$
4. $(-2)^5 = -32$. The expression $-2^5 = -(2^5) = -32$ (Both are the same sign here because exponent is odd; for even exponents the sign differs.)

Conclusion

Exponents give a powerful shorthand for repeated multiplication. Mastering the rules above makes simplifying algebraic expressions and working with scientific notation much easier. If you want, I can create more worked examples, visual explanations, or practice sets at different difficulty levels.