Year 8 Mathematics: Applying Algebra and the Index Laws

Comprehensive vocabulary flashcards reviewing essential algebraic modelling, linear cost expressions, the three index laws, and extension concepts from Week 21.

Base

In the power expression $a^n$, the number or variable $a$ that is multiplied by itself repeatedly.

Index

In the power expression $a^n$, the number or variable $n$ that specifies how many copies of the base are multiplied together.

Index Law 1 (Multiplying Powers)

The rule stating that when multiplying powers with the same base, add the indices: $a^m \times a^n = a^{m+n}$.

Index Law 2 (Dividing Powers)

The rule stating that when dividing powers with the same base, subtract the bottom index from the top index: $a^m \div a^n = \frac{a^m}{a^n} = a^{m-n}$.

Index Law 3 (Power of a Power)

The rule stating that when raising a power to another power, multiply the indices: $(a^m)^n = a^{mn}$.

Zero Index Rule

The mathematical law stating that $a^0 = 1$ for every base except $0$, which arises from dividing equal powers ($a^n \div a^n = a^{n-n} = a^0 = 1$).

Same-Base Condition

The fundamental requirement that index laws apply only when powers share identical bases; expressions with different bases (such as $2^3 \times 3^4$) cannot be combined or simplified using index laws.

Index Form

Writing a mathematical result as a base raised to a power (such as $3^5$) rather than evaluating it into an ordinary numerical value (such as $243$).

Equivalent Expressions

Two or more algebraic expressions that produce the exact same numerical result for every possible value of the variable.

Fee Paid Once (Fixed Charge)

In a linear cost situation, a constant charge that does not depend on the variable and is written as the standalone term with no variable attached.

Charge for Each One (Variable Rate)

In a linear cost situation, a recurring unit rate that is multiplied by the variable representing the number of units or hours (for example, $\$5$ per ride for $n$ rides is written as $5n$).

Power on a Bracket

An exponent placed outside parentheses that applies to every term and factor inside, coefficients included (for example, $(3x)^4 = 3^4 x^4$).

Multiplying Coefficients with Powers

The rule that numerical coefficients in front of algebraic powers are multiplied together as regular numbers, while the powers themselves are combined by adding indices (for example, $4m^2 \times 5m^3 = 20m^5$).

Scope of a Zero Index

The principle that an exponent of zero applies only to the exact factor it is immediately attached to (for example, $4x^0 = 4 \times 1 = 4$, whereas $(4x)^0 = 1$).

Lone Variable Index Convention

The algebraic rule that any variable or number written on its own without a visible exponent has an implicit index of $1$ ($a = a^1$).

Negative Index

A power with a negative exponent that denotes the reciprocal of the base raised to the positive exponent: $a^{-n} = \frac{1}{a^n}$ (where $a \neq 0$).

Scientific Notation

A standard method of writing numbers in the form $a \times 10^n$, where $1 \le a < 10$ and $n$ is an integer.

Exponential Equation

An equation in which the variable is in the exponent; when both sides have matching bases ($a^x = a^y$), their indices must be equal ($x = y$).

Exponential Growth

A rate of increase where a quantity is repeatedly multiplied by a constant factor across successive units of time (such as an initial amount of $7$ doubling every hour, modelled by $7 \times 2^n$ after $n$ hours).

Standard Algebraic Conventions

The rules specifying to write coefficients before variables ($3x$, never $x3$), list variables in alphabetical order, use exponents rather than repeated letters ($x^2$, never $xx$), and omit a coefficient of $1$.