College Algebra: Exponents, Polynomials, and Scientific Notation Study Guide

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: $a^m \cdot a^n = a^{m+n}$

• Example: $x^3 \cdot x^5 = x^{3+5} = x^8$

Quotient Rule for Exponents

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

• Rule: $\frac{a^m}{a^n} = a^{m-n}$, where $a \neq 0$

• Example: $\frac{y^7}{y^3} = y^{7-3} = y^4$

Power Rule for Exponents

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

• Rule: $(a^m)^n = a^{mn}$

• Example: $(x^2)^4 = x^{2 \times 4} = x^8$

Power of a Product and Power of a Quotient

• Power of a Product: $(ab)^n = a^n b^n$

• Power of a Quotient: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$, $b \neq 0$

• Example: $\left(\frac{x^2 y^3}{z^4}\right)^2 = \frac{x^{4} y^{6}}{z^{8}}$

Negative and Zero Exponents

• Negative Exponent: $a^{-n} = \frac{1}{a^n}$, $a \neq 0$

• Zero Exponent: $a^0 = 1$, $a \neq 0$

• Example: $x^{-3} = \frac{1}{x^3}$

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: $a \times 10^n$, where $1 \leq a < 10$ and $n$ is an integer

• Example: $5,870,000,000 = 5.87 \times 10^9$

Converting Between Scientific Notation and Standard Form

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

• Example: $2.5 \times 10^8 = 250,000,000$

Operations with Scientific Notation

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

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

• Example: $(7.16 \times 10^4) \times (2.3 \times 10^3) = (7.16 \times 2.3) \times 10^{4+3} = 16.468 \times 10^7 = 1.6468 \times 10^8$

Polynomials

Definition and Terminology

A polynomial is an algebraic expression consisting of terms in the form $ax^n$, where $a$ is a coefficient and $n$ 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., $x$, $y$)

• Constant: A term without a variable

Classifying Polynomials

• Monomial: 1 term (e.g., $5x^3$)

• Binomial: 2 terms (e.g., $x^2 - 4$)

• Trinomial: 3 terms (e.g., $x^2 + 2x + 1$)

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: $(2x^2 + 3x) + (x^2 - x) = 3x^2 + 2x$

Table: Parts of a Polynomial

Polynomial	Terms	Coefficients	Variables	Constants
216x - 1	2	216, -1	x	None

Applications: Area and Geometry

Area Formulas

• Square: $\text{Area} = \text{side}^2$

• Rectangle: $\text{Area} = \text{length} \times \text{width}$

• Triangle: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$

Example: Area Calculation

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

• Example: If a triangle has base $2.8$ cm and height $3.4$ cm, $\text{Area} = \frac{1}{2} \times 2.8 \times 3.4 = 4.76$ cm$^2$

Applications: Compound Interest

Compound Interest Formula

• Formula: $A = P \left(1 + \frac{r}{n}\right)^{nt}$

• Where:

  • $A$ = amount after time $t$

  • $P$ = principal (initial amount)

  • $r$ = annual interest rate (decimal)

  • $n$ = number of times compounded per year

  • $t$ = number of years

• Example: $P = 5000$, $r = 0.04$, $n = 12$, $t = 5$ $A = 5000 \left(1 + \frac{0.04}{12}\right)^{12 \times 5}$

Scientific Constants: Avogadro's Number

Definition

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

• Value: $6.022 \times 10^{23}$

Summary Table: Key Exponent Rules

Rule	Formula	Example
Product Rule	$a^m \cdot a^n = a^{m+n}$	$x^2 \cdot x^3 = x^5$
Quotient Rule	$\frac{a^m}{a^n} = a^{m-n}$	$y^5 / y^2 = y^3$
Power Rule	$(a^m)^n = a^{mn}$	$(x^2)^3 = x^6$
Negative Exponent	$a^{-n} = \frac{1}{a^n}$	$x^{-4} = 1/x^4$
Zero Exponent	$a^0 = 1$	$5^0 = 1$

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.