Exponents and Their Properties

Product Rule for Exponents

The product rule states that when multiplying two expressions with the same base, add the exponents.

• Rule:

• Example:

Quotient Rule for Exponents

The quotient rule states that when dividing two expressions with the same base, subtract the exponents.

• Rule: , where

• Example:

Power Rule for Exponents

The power rule states that when raising a power to another power, multiply the exponents.

• Rule:

• Example:

Power of a Product and Power of a Quotient

• Power of a Product:

• Power of a Quotient: ,

• Example:

Negative and Zero Exponents

• Negative Exponent: ,

• Zero Exponent: ,

• Example:

Scientific Notation and Standard Form

Writing Numbers in Scientific Notation

Scientific notation expresses numbers as a product of a number between 1 and 10 and a power of 10.

• Form: , where and is an integer

• Example:

Converting Between Scientific Notation and Standard Form

• To convert from scientific notation to standard form, move the decimal point places to the right (if is positive) or to the left (if is negative).

• Example:

Operations with Scientific Notation

• Multiplication: Multiply the decimal parts and add the exponents.

• Division: Divide the decimal parts and subtract the exponents.

• Example:

Polynomials

Definition and Terminology

A polynomial is an algebraic expression consisting of terms in the form , where is a coefficient and is a non-negative integer.

• Term: Each part of a polynomial separated by + or -

• Coefficient: The numerical factor of a term

• Degree: The highest exponent of the variable in the polynomial

• Variables: The letters representing unknowns (e.g., , )

• Constant: A term without a variable

Classifying Polynomials

• Monomial: 1 term (e.g., )

• Binomial: 2 terms (e.g., )

• Trinomial: 3 terms (e.g., )

Operations with Polynomials

• Addition/Subtraction: Combine like terms.

• Multiplication: Use distributive property or FOIL for binomials.

• Division: Divide each term in the numerator by the denominator.

• Example:

Table: Parts of a Polynomial

Applications: Area and Geometry

Area Formulas

• Square:

• Rectangle:

• Triangle:

Example: Area Calculation

• Given side or base and height, substitute values into the appropriate formula and solve.

• Example: If a triangle has base cm and height cm, cm

Applications: Compound Interest

Compound Interest Formula

• Formula:

• Where:

  • = amount after time

  • = principal (initial amount)

  • = annual interest rate (decimal)

  • = number of times compounded per year

  • = number of years

• Example: , , ,

Scientific Constants: Avogadro's Number

Definition

• Avogadro's Number: The number of atoms or molecules in one mole of a substance.

• Value:

Summary Table: Key Exponent Rules

Additional info:

• Some problems involve evaluating expressions without a calculator, reinforcing mental math and exponent rules.

• Geometry applications connect algebraic manipulation to real-world contexts.

• Scientific notation and significant figures are essential for handling very large or small numbers in science and engineering.