1
Evaluate without ambiguity
Apply the order of operations to an expression with nested brackets, exponents and a fraction bar, and get the value a marker gets.
Course · Algebra
Foundations of Algebra
Arithmetic, made general. Why a letter can stand for a number, which rearrangements are always allowed and which only look allowed, and the exponent and radical rules that everything after this course leans on.
a(b + c) = ab + ac the distributive law
x^m · x^n = x^(m+n) exponents add
x^(1/n) = the n-th root of x
|x| = x if x ≥ 0, −x if x < 0
What this course is about
- Subject
- Algebra
- Lessons
- 13
- Format
- Interactive
Overview
What you will be able to do
By the end you can read, evaluate and simplify the real-number expressions the next courses use, with a stated reason for every rearrangement.
2
Name the property you used
Say which of commutativity, associativity, distribution or the identity you just applied — and notice when you have used one that does not exist.
3
Simplify powers and radicals under their conditions
Combine powers with a common base, handle zero and negative exponents, convert between x^(1/n) and roots, decide whether a real root exists, and simplify a numerical radical exactly.
4
Turn a sentence into an expression
Translate an English quantity into algebra, keeping track of what the letter stands for — which is where most word-problem errors are made.
Lessons
13 lessons
“Real Numbers and the Number Line” and “Order of Operations” settle the numbers and how notation is read; “Variables, Expressions, and Terms” names the parts of an expression before “Properties of the Real Numbers” licenses rearrangements. Absolute value, exponents and radicals come next; expanding, collecting, evaluating and translating expressions close the course.
Real Numbers and the Number LineWhich numbers there are, and where each one sits.Not marked
Order of OperationsOne expression, one value, and the convention that guarantees it.Not marked
Variables, Expressions, and TermsVariables, terms, coefficients, and the anatomy of an expression.Not marked
Properties of the Real NumbersCommutativity, associativity, distribution, identities, inverses.Not marked
Absolute ValueDistance from zero, and why the definition has two cases.Not marked
Integer ExponentsThe laws, including what a zero and a negative exponent mean.Not marked
Scientific NotationWriting very large and very small numbers, and computing with them.Not marked
Roots and RadicalsSquare roots, n-th roots, and which of them are real.Not marked
Rational ExponentsWhy x^(1/n) must mean the n-th root if the laws are to hold.Not marked
The Distributive LawThe one property that connects addition and multiplication.Not marked
Combining Like TermsWhat makes two terms alike, and what does not.Not marked
Evaluating ExpressionsSubstitution done carefully, brackets and all.Not marked
Translating Words into AlgebraTurning a sentence into an expression, and naming the unknown.Not marked
Recommended background
Fractions, negatives, and long division.
Practice suggestions
Do the arithmetic yourself before you read the answer. Every lab shows its steps, and the steps are worth more than the result — the result you could have got from a calculator.
Take the property lessons seriously even if the arithmetic is easy. “Properties of the Real Numbers” is the reason -(x - 3) is -x + 3, and that single sign is the most common error in the next two courses.
When a lab disagrees with you, find which step differs rather than which answer. The step is the thing you will repeat a thousand times.
What it does not cover
Proof of the field axioms. This course states the properties of the real numbers and uses them; constructing the reals from the rationals belongs to analysis.
Complex numbers. sqrt(-4) has no value on this course and is said to have none; Quadratics and Complex Numbers introduces i and gives it one.
Trigonometry and logarithms. Logarithms arrive in Exponential and Logarithmic Functions, where exponentials make them necessary; trigonometry is not on this path at all.