Algebraic Techniques and Index Laws

Vocabulary flashcards covering the key terms of algebraic techniques, robot/AI applications in mathematics, and index laws from Chapter 5 of the Stage 4 curriculum.

Natural language processing

A technology used by chatbots to enable human-to-computer communication.

ROSA (Robotic Surgical Assistant)

A medical robot that combines computer navigation, 3D modelling, and a robotic arm to allow surgeons to operate with increased precision.

Pronumeral

A letter used to represent one or more numbers, also referred to as a variable.

Expression

A combination of numbers and pronumerals combined with mathematical operations, such as $3x + 2yz$.

Term

A part of an expression consisting only of pronumerals, numbers, multiplication, and division, such as $9a$, $10cd$, or $\frac{3x}{5}$.

Coefficient

The number in front of a pronumeral; if the term is subtracted, it is negative, and if no number is present, it is $1$.

Constant term

A term in an algebraic expression that does not contain any variables.

Substitution (Evaluation)

The process of replacing each pronumeral in an expression with a number to obtain a final value.

Equivalent expressions

Expressions that have equal values regardless of which number is substituted for each pronumeral.

Commutative laws

Arithmetic properties stating that $a + b = b + a$ and $a \times b = b \times a$ for all values of $a$ and $b$.

Associative laws

Arithmetic properties stating that $a + (b + c) = (a + b) + c$ and $a \times (b \times c) = (a \times b) \times c$ for all values of $a$, $b$, and $c$.

Like terms

Terms that contain exactly the same pronumerals with the same powers, although the order of pronumerals may differ (e.g., $4ab$ and $7ba$).

Algebraic fraction

An expression involving division that may include algebraic expressions in the numerator or the denominator.

Lowest Common Denominator (LCD)

The smallest multiple of the denominators of two or more algebraic fractions, required for addition and subtraction.

Reciprocal

The inverse of an algebraic fraction formed by swapping the numerator and denominator; e.g., the reciprocal of $\frac{3b}{4}$ is $\frac{4}{3b}$.

Distributive law

A mathematical rule used to rewrite an expression without brackets, given as $a(b + c) = ab + ac$.

Expanding brackets

The process of using the distributive law to convert an expression from a product form (with brackets) into a sum or difference form.

Factorising

The reverse procedure of expanding, aimed at writing an algebraic expression as the product of two or more factors.

Highest Common Factor (HCF)

The largest factor that divides into each term of a set, used outside the brackets during factorisation.

Index notation

A convenient way to describe repeated multiplication using a base and an index, such as $a^n$.

Base (Indices)

The number or pronumeral that is being multiplied by itself in index notation; in the expression $a^n$, this is the $a$.

Index (Exponent or Power)

In index notation $a^n$, the number $n$ that indicates how many copies of the base are being multiplied.

Index law for multiplication

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

Index law for division

The rule stating that when dividing powers with the same base, the indices are subtracted: $a^m \div a^n = a^{m-n}$.

Zero index law

The principle that any number (other than $0$) raised to the power of $0$ results in $1$ ($a^0 = 1$).

Power of a power

The index law stating that to raise a power to another power, the indices are multiplied: $(a^m)^n = a^{mn}$.

Scientific notation

A method used to express very large or very small numbers using indices, such as writing $8\,000\,000$ as $8 \times 10^6$.