Properties of Exponents

Flashcards covering the fundamental properties of exponents with rules and worked examples from lecture notes.

Product Property (multiplying)

Rule: $x^a \cdot x^b = x^{a+b}$. Example: $2^3 \cdot 2^4 = 2^{3+4} = 2^7$.

Quotient Property (Dividing)

Rule: $\frac{x^a}{x^b} = x^{a-b}$. Example: $\frac{5^6}{5^2} = 5^{6-2} = 5^4$.

Power of a Power Property

Rule: $(x^a)^b = x^{a \cdot b}$. Example: $(3^3)^4 = 3^{3 \cdot 4} = 3^{12}$.

Power of a Product Property

Rule: $(xy)^a = x^a \cdot y^a$. Example: $(2x)^3 = 2^3 \cdot x^3 = 8x^3$.

Power of a Quotient Property

Rule: $\left(\frac{x}{y}\right)^a = \frac{x^a}{y^a}$. Example: $\left(\frac{3}{4}\right)^2 = \frac{3^2}{4^2} = \frac{9}{16}$.

Zero Exponent Property

Rule: $x^0 = 1$. Example: $999^0 = 1$.

Negative Exponent Property

Rule: $x^{-a} = \frac{1}{x^a}$. Example: $4^{-2} = \frac{1}{4^2} = \frac{1}{16}$.

Product Property of Exponents

A rule stating that when multiplying expressions with the same base, you keep the base and add the exponents: $x^a \cdot x^b = x^{a+b}$

Product Property (multiplying)

Rule: $x^a \cdot x^b = x^{a+b}$. Example: $2^3 \cdot 2^4 = 2^{3+4} = 2^7$.

Quotient Property (Dividing)

Rule: $\frac{x^a}{x^b} = x^{a-b}$. Example: $\frac{5^6}{5^2} = 5^{6-2} = 5^4$.

Power of a Power Property

Rule: $(x^a)^b = x^{a \cdot b}$. Example: $(3^3)^4 = 3^{3 \cdot 4} = 3^{12}$.

Power of a Product Property

Rule: $(xy)^a = x^a \cdot y^a$. Example: $(2x)^3 = 2^3 \cdot x^3 = 8x^3$.

Power of a Quotient Property

Rule: $\left(\frac{x}{y}\right)^a = \frac{x^a}{y^a}$. Example: $\left(\frac{3}{4}\right)^2 = \frac{3^2}{4^2} = \frac{9}{16}$.

Zero Exponent Property

Rule: $x^0 = 1$. Example: $999^0 = 1$.

Negative Exponent Property

Rule: $x^{-a} = \frac{1}{x^a}$. Example: $4^{-2} = \frac{1}{4^2} = \frac{1}{16}$.

Product Property of Exponents

A rule stating that when multiplying expressions with the same base, you keep the base and add the exponents: $x^a \cdot x^b = x^{a+b}$.

Real Numbers

The set containing all rational and irrational numbers.

Irrational Numbers

Decimals that never terminate and have no repetition, such as $\sqrt{3}$, $\pi$, $\sqrt{5}$, or $\sqrt{41}$.

Rational Numbers

Numbers that can be written as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers, represented by terminating or repeating decimals.

Integers

The set of whole numbers and their opposites: ${…, -3, -2, -1, 0, 1, 2, 3, …}$.

Whole Numbers

The set of non-negative integers starting from zero: ${0, 1, 2, 3, …}$.

Natural Numbers

The set of positive counting numbers: ${1, 2, 3, 4, …}$.

Classification of $\sqrt{16}$

Rational number, integer, whole number, and natural number, because $\sqrt{16} = 4$.

Classification of $-3.4$

Rational number, because it is a terminating decimal.

Classification of $0$

Rational number, integer, and whole number (excluding natural numbers).