Exponents and Their Properties

Product Rule for Exponents

The product rule allows us to multiply powers with the same base by adding their exponents.

• Definition:

• Example:

Quotient Rule for Exponents

The quotient rule allows us to divide powers with the same base by subtracting the exponents.

• Definition:

• Example:

Power Rule for Exponents

The power rule is used when raising a power to another power; multiply the exponents.

• Definition:

• Example:

Power of a Product and Power of a Quotient

These rules distribute the exponent to each factor or term inside the parentheses.

• Power of a Product:

• Power of a Quotient:

• Example:

Zero and Negative Exponents

Zero exponents result in 1 (for nonzero bases), and negative exponents represent reciprocals.

• Zero Exponent: (for )

• Negative 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.

• Format: where and is an integer

• Example:

Converting Between Scientific Notation and Standard Form

• To convert to standard form, multiply the decimal by the power of ten.

• To convert to scientific notation, move the decimal point to create a number between 1 and 10, counting the moves as the exponent.

• Example:

Operations with Scientific Notation

• When multiplying, multiply the decimal parts and add the exponents.

• When dividing, divide the decimal parts and subtract the exponents.

• Example:

Polynomials: Structure and Operations

Definition and Classification

A polynomial is an expression consisting of variables, coefficients, and non-negative integer exponents.

• Term: Each part separated by + or -

• Degree: The highest exponent of the variable

• Coefficient: The numerical factor of a term

• Example: has three terms, degree 2, coefficients 5, -3, and 7

Operations with Polynomials

• Addition/Subtraction: Combine like terms (same variable and exponent)

• Multiplication: Use distributive property or FOIL for binomials

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

• Example:

Polynomial Table Example

Polynomials can be analyzed by breaking them into terms, coefficients, variables, and constants.

Geometry Applications in Algebra

Area Formulas

Algebraic expressions are used to represent geometric quantities such as area.

• Square:

• Rectangle:

• Triangle:

Using Formulas in Word Problems

• Substitute given values into the formula to solve for area or other quantities.

• Example: If a triangle has base 2.4 cm and height 3.6 cm, cm2

Applications: Compound Interest Formula

Compound Interest

The compound interest formula calculates the amount in an account after a certain time with interest compounded periodically.

• Formula:

• Where:

  • = amount after time

  • = principal (initial amount)

  • = annual interest rate (decimal)

  • = number of times interest is compounded per year

  • = number of years

• Example: , , ,

Additional Info

• Avogadro's number is a constant used in chemistry, but its representation in scientific notation is a useful algebraic skill:

• Some problems involve identifying the degree and type of polynomial, which is foundational for later topics such as factoring and graphing.