Lesson 1: Algebra

Distributive Law

A rule that allows multiplication to be distributed across terms inside parentheses.

Distributive Law Formula

a(b + c) = ab + ac.

Distributive Law for Subtraction

a(b − c) = ab − ac.

Distributive Law Example

4(x + 3) = 4x + 12.

Identity Law

Adding zero to any number gives the same number, while multiplying any number by one gives the same number.

Identity Law Formula

a + 0 = a and a × 1 = a.

Identity Law Example

14 + 0 = 14 and 14 × 1 = 14.

Closure Law

If two natural numbers are added or multiplied, the result is always a natural number.

Closure Law Addition

a + b ∈ N.

Closure Law Multiplication

a × b ∈ N.

Closure Law Example

12 + 9 = 21 and 8 × 7 = 56, and both results are natural numbers.

∈ Symbol

Means "is an element of" or "belongs to."

Law of Exponents

Rules used to simplify expressions involving repeated multiplication and powers.

Product Rule of Exponents

When multiplying powers with the same base, keep the base and add the exponents.

Product Rule Formula

aᵐ × aⁿ = aᵐ⁺ⁿ.

Product Rule Condition

The expressions must have the same base.

Quotient Rule of Exponents

When dividing powers with the same base, keep the base and subtract the exponents.

Quotient Rule Formula

aᵐ ÷ aⁿ = aᵐ⁻ⁿ.

Quotient Rule Condition

The base must be the same and a ≠ 0.

Power of a Product Rule

When a product inside parentheses is raised to an exponent, raise each factor to that exponent.

Power of a Product Formula

(ab)ⁿ = aⁿbⁿ.

Power of a Quotient Rule

When a fraction inside parentheses is raised to an exponent, raise both the numerator and denominator to that exponent.

Power of a Quotient Formula

(a/b)ⁿ = aⁿ/bⁿ.

Zero Exponent Rule

Any nonzero base raised to the zero power equals 1.

Zero Exponent Formula

a⁰ = 1, provided a ≠ 0.

Negative Exponent Rule

A negative exponent means the base is moved to the opposite side of the fraction bar and the exponent becomes positive.

Negative Exponent in Numerator

If the base has a negative exponent in the numerator, move it to the denominator and make the exponent positive.

Negative Exponent in Denominator

If the base has a negative exponent in the denominator, move it to the numerator and make the exponent positive.