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Exponential Notation in Algebra

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Summary of Exponent Properties

When $a$ and $b$ are real numbers and $m$ and $n$ are rational numbers, the following seven properties govern how exponents behave across multiplication, division, and raising to a power:

Property	Rule
Product Property	$a^m \cdot a^n = a^{m+n}$
Power Property	$(a^m)^n = a^{m \cdot n}$
Product to a Power	$(ab)^m = a^m b^m$
Quotient Property ( $m > n$ )	$\frac{a^m}{a^n} = a^{m-n}$ , $a \neq 0$
Quotient Property ( $n > m$ )	$\frac{a^m}{a^n} = \frac{1}{a^{n-m}}$ , $a \neq 0$
Zero Exponent	$a^0 = 1$ , $a \neq 0$
Quotient to a Power	$\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$ , $b \neq 0$

These properties were originally established for whole-number exponents and later extended to integer exponents (including zero and negative values). With the introduction of rational exponents, the same rules apply when the exponents are fractions — the arithmetic operations on the exponents (adding, subtracting, multiplying) simply involve fraction arithmetic instead of integer arithmetic. The first three properties handle products and powers of products, while the last four handle quotients, the zero-exponent special case, and powers of quotients. Together they form a complete toolkit for simplifying any expression built from powers of the same base, whether those powers are whole numbers, integers, or rational numbers.

Updated 2026-05-01

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