Exponent Laws and Powers

Vocabulary flashcards covering the components of powers and foundational exponent laws, including product, quotient, power of a power, power of a product/quotient, zero exponent, and negative exponent rules.

Power

A mathematical expression representing repeated multiplication that consists of both a base and an exponent (for example, $10^9$).

Base

The number or variable in a power that is repeatedly multiplied by itself (for example, $10$ in the power $10^9$).

Exponent

The number or variable in a power that indicates how many times the base is multiplied by itself as a factor (for example, $9$ in the power $10^9$).

Exponent Law for a Product of Powers

To multiply powers with the same base, add the exponents: $a^m \times a^n = a^{m+n}$, where variable $a$ is any integer except $0$, and variables $m$ and $n$ are any whole numbers.

Exponent Law for a Quotient of Powers

To divide powers with the same base, subtract the exponents: $a^m \div a^n = a^{m-n}$ ($m \ge n$), where $a$ is any integer except $0$, and $m$ and $n$ are any whole numbers.

Exponent Law for a Power of a Power

To raise a power to a power, multiply the exponents: $(a^m)^n = a^{mn}$, where $a$ is any integer except $0$, and $m$ and $n$ are any whole numbers.

Exponent Law for a Power of a Product

To raise a product to a power, apply the exponent to each factor in the product: $(ab)^m = a^m b^m$, where $a$ and $b$ are any integers except $0$, and $m$ is any whole number.

Exponent Law for a Power of a Quotient

To raise a quotient to a power, apply the exponent to both the numerator and the denominator: $(\frac{a}{b})^n = \frac{a^n}{b^n}$ ($b \ne 0$), where $a$ and $b$ are any integers except $0$, and $n$ is any whole number.

Zero Exponent Rule

Any non-zero base raised to the exponent zero is equal to one: $x^0 = 1$ (or $a^0 = 1$, where $a \ne 0$; for example, $7^0 = 1$).

Exponent of One Rule

Any base raised to the power of one is equal to the base itself: $x^1 = x$ (or $a^1 = a$; for example, $6^1 = 6$).

Negative Exponent Rule

A base raised to a negative exponent equals the reciprocal of the base raised to the opposite positive exponent: $x^{-n} = \frac{1}{x^n}$ and $x^{-1} = \frac{1}{x}$ (for example, $4^{-1} = \frac{1}{4}$ and $x^{-3} = \frac{1}{x^3}$).