Introduction to Exponent Rules and Polynomials

Flashcards covering basic exponent properties, parity, exponential notation, and the rules for polynomial multiplication and negative exponents.

Exponential notation

A mathematical format that requires a single base raised to a single exponent.

Parity

The state of an exponent being either odd or even, which determines the sign of the result when a negative number is raised to that power.

Negative Base sign without Parentheses

In expressions like $-2^2$, the base is only $2$, meaning the exponent does not affect the negative sign, resulting in $-1 \times 2^2 = -4$.

Negative Base sign with Parentheses

In expressions like $(-2)^2$, the base is $-2$, meaning the exponent affects the negative sign, resulting in a positive outcome for even exponents.

Zero Exponent Rule

Any non-zero value $a$ raised to the power of zero is always equal to $1$: $a^0 = 1$.

Degree of a Constant

The degree of any constant is zero because it can be expressed with a variable raised to the zero power, such as $15x^0$.

Product Rule for Exponents

When multiplying two numbers with the exact same base, the exponents are added while keeping the base the same: $a^m \times a^n = a^{m+n}$.

Quotient Rule for Exponents

When dividing two numbers with the same base, the exponent of the denominator is subtracted from the exponent of the numerator: $\frac{a^m}{a^n} = a^{m-n}$.

Power to a Power Rule

When an exponent is raised to another exponent, the two exponents are multiplied: $(a^m)^n = a^{m \times n}$.

Power of a Product Rule

If multiplication occurs inside a base raised to an exponent, the exponent can be distributed to each factor: $(ab)^m = a^m b^m$.

Power of a Quotient Rule

When a fraction is raised to an exponent, the exponent is distributed to both the numerator and the denominator: $(\frac{a}{b})^m = \frac{a^m}{b^m}$.

Negative Exponent Rule

A negative exponent indicates that the base belongs on the other side of the fraction bar to make the exponent positive: $a^{-n} = \frac{1}{a^n}$.

Reciprocal Rule for Fractions

A fraction raised to a negative power can be made positive by taking the reciprocal of the fraction base: $(\frac{a}{b})^{-n} = (\frac{b}{a})^n$.

Order of Operations with Exponents

Exponents must be evaluated or simplified before performing multiplication or distribution.

Function Product Notation $fg(x)$

The notation for the product of two functions, defined as the value of the first function multiplied by the second function: $f(x) \times g(x)$.