Fundamentals of Algebra

Explore the types of numbers from natural to real numbers, including integers, rationals, irrationals, and constants like pi, and learn how the number line and decimals distinguish these categories.

Explore why dividing by zero is undefined, illustrated by a $10 loan repaid in two installments over five days and a $200 loan with $0 daily payments.

Master the identity law by seeing zero as the additive identity and one as the multiplicative identity, since adding zero or multiplying by one leaves numbers unchanged.

Explore the commutative laws that let you swap addends or factors in addition or multiplication, such as 12 percentage of 50; division is not commutative.

Master the rules of multiplying negative numbers: same signs yield positive, opposite signs yield negative; use the commutative property and even/odd negatives to simplify.

Explore the inverse properties of addition and multiplication, including additive and multiplicative inverses, with examples of reversing operations and the special case of zero.

Learn four ways to write multiplication—times sign, dot, parentheses, and adjacency—and apply the order of operations, multiplying before adding or subtracting, with notes on algebra adjacency rules.

Apply the order of operations (BEDMAS) to calculations by following brackets first, exponents, then multiply and divide left to right, and finally add or subtract.

Learn how absolute value measures distance from zero on the number line, removing negative signs. Use examples like |-3|=3 and |-18|=18 to solve distance problems.

Learn the associative laws for addition and multiplication by grouping numbers with parentheses first, see examples like 18 + 23 + 7 and 2 × 23 × 5, with practice.

Explore the distributive law by showing how a(b+c)=ab+ac with examples like 3(2+4)=3×2+3×4, and apply it to simplify multiplication and subtraction.

Explore equal and rational numbers by multiplying or dividing the numerator and denominator by the same number, simplifying fractions while keeping value constant.

Learn how to express any rational number in standard form by canceling common factors, transforming to a reduced fraction like 1/10, and noting when cancellation ends.

Multiply and divide rational numbers by simplifying to standard form, canceling factors, and using reciprocals. Master handling complex fractions with keep it, change it, flip it, and multiply across.

Learn to add and subtract rational numbers, first with same denominators by keeping the common denominator, then with different denominators, and simplify to standard form.

Learn to add and subtract rational numbers with different denominators using cross multiplication and the lcm method, including steps, simplifying, and handling signs.

apply the Elysium method to add and subtract rational numbers by finding the least common multiple using a division ladder, convert to common denominators, then combine the numerators.

Explore exponents as a shorthand for repeated multiplication, using base and power with examples like four to the second power and three to the fourth power, noting negative bracket rules.

Explore why any non-zero number raised to the zero power equals one, using doubling oranges to illustrate exponential form and consistent notation.

Explains why zero to the zero is undefined and why zero cannot be treated as any number. It shows that nonzero bases raised to zero power equal one.

Visualize square numbers as blocks to reveal shortcuts for subtracting consecutive squares and summing odd numbers, with sums equal to n^2 and n^2+n.

Explore how square numbers arise by multiplying a number by itself, as in nine from nine squares. Learn that squaring, the power of two, means a number times itself—five squared.

Discover the square root concept, identifying square numbers like 16 and recognizing positive and negative roots. Explore the radical symbol and radicand, and practice finding roots such as 25.

Learn the prime factorization method to find square roots without a calculator by reducing numbers to prime factors and forming pairs to extract the root.

Practice prime factorization to find square roots, as shown by factoring 9025 into 5, 5, 19, 19 and deriving the root 95; upcoming videos will reveal faster factorization methods.

Learn to use the long division method to find square roots by grouping digits in pairs, selecting the closest square, and applying trial and error.

Learn to simplify square roots of products by factoring into perfect squares and pulling them out, applying the product rule of square roots with examples like sqrt2 and 2 sqrt3.

Apply the quotient rule to simplify radicals in fractions, check for perfect squares, and use the product rule to rewrite under a square root, with examples like sqrt(152/90) and sqrt(312/13).

Explore the product rule for exponents, add powers on like bases, handle negative bases, and use prime factorization to express numbers in exponential form.

Learn the quotient rule for dividing exponential expressions with the same base, subtract exponents, and move the denominator to the numerator, reversing the exponent sign and handling negative bases.

Learn how exponents with different bases and the same powers combine: multiply bases as (A*B)^n or divide bases as (A/B)^n, then apply the shared exponent.

Explore the power rule and product rule for exponents, learn how to combine powers with the same base by multiplying exponents, and apply these rules to simplify expressions.

Apply the laws of exponents to simplify expressions by distributing powers inside brackets, multiplying exponents for power-to-power cases, combining coefficients and bases, and treating bare variables as exponent one.

Rewrite expressions with positive exponents to simplest form using reciprocal and multiplicative inverse, bring negative exponents to the numerator, and apply power and product rules.

Rewrite negative exponents as positive, move terms to the appropriate numerator or denominator, then simplify by canceling common factors and combining like bases and exponents.

Discover how to write and compute with very large or very small numbers using scientific notation, including the form c × 10^n, positive and negative exponents, and basic product rules.

Learn how algebra uses letters as variables to form expressions by combining numbers and variables with addition, subtraction, multiplication, and division, and identify terms, coefficients, and constants.

Identify like terms by grouping terms with the same variable and powers. Recognize constants also as like terms, since their power is zero.

Group like terms and distribute to simplify expressions. Add or subtract polynomials by combining coefficients with the same variables.

Learn to multiply algebraic expressions by distributing each term across the other, handle signs, multiply numbers, and add exponents for like bases, noting that multiplication does not combine like terms.

Explore polynomial expressions and their types—monomials, binomials, and trinomials—demonstrating how to form polynomials, determine degree, and write in standard form with non-negative exponents.

Master the difference of two squares by using the distributive property to multiply binomials in the form a+b and a-b, enabling quicker manipulation of algebra.

Explore the geometrical proof of the square of a binomial with positive sign, showing that (A+B)^2 equals A^2 + 2AB + B^2.

Explore a geometric proof of the binomial square (a−b)², showing a²−2ab+b² by dividing a square into four shapes and summing their areas.

Master shortcuts for difference of squares and binomial squares using the distributive property; multiply first and last terms, cancel inner and outer terms, and obtain middle term as two ab.

Identify sum and difference patterns and binomial squares, using brackets to form (a+b)^2 or (a−b)^2 and determine the middle term by doubling the product.

Apply the difference of two squares to factor binomials like (2x+3)(2x-3) and simplify expressions such as (37^2-24^2)/(37+24), revealing the simplified result.

Treat algebra as a language by translating between plain English and math, isolate the variable, balance equations, and solve for x using addition, subtraction, multiplication, and division.

Apply the transpose method to solve equations by moving terms across the equals sign, turning plus to minus and multiplication into division as needed.

Explore how equations express equality and how formulas relate variables, and rearrange a formula to make a new variable the subject, using the volume formula v = l w h.

Learn to solve linear equations with one variable by simplifying each side using the order of operations, isolating variable terms on one side, and applying multiplication or division.

Master solving linear equations with one variable by distributing, combining like terms, isolating the variable, and applying division or cross multiplication to find x or y.

Learn to solve linear equations with one variable by isolating the variable, using product and quotient rules for exponents, and equating powers when bases match.

Learn to cross multiply to solve equations and proportions, convert A/B = C/D to cross-multiplied form, cancel common denominators, and isolate x by distributing and dividing by coefficients.

Explore the coordinate plane, a two-dimensional surface with x and y axes, and locate points using order pairs, starting from the origin and identifying quadrants by positive and negative coordinates.

Solve for y, plot at least two points that satisfy the equation, and connect them to form a straight line extended indefinitely. See that every point on that line is a solution, while points off it are not.

Isolate y to get y = 2 minus 3x for the line 3x + y = 2. Plot the points (-1, 5) and (1, -1), then connect them to form the graph.

Compute slope as the rise over run using two points on a coordinate plane, then apply the slope-intercept form y = mx + b.

Discover how slope values measure vertical change relative to horizontal change, distinguishing positive, negative, zero, and undefined slopes.

apply the slope formula to two points, compute m = (y2 - y1)/(x2 - x1) with (-3,4) and (5,2), showing a negative slope of -1/4.

Learn to graph linear equations in slope-intercept form by plotting the y-intercept and applying rise-over-run from the slope, with a step-by-step example of y = 2x + 1.

Convert the equation to slope-intercept form y = mx + b, identify m and b, and graph by plotting the y-intercept and using rise over run.