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

Exponents and Their Properties

Product Rule for Exponents

The product rule allows you to multiply expressions with the same base by adding their 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 allows you to divide expressions with the same base by subtracting 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 is used 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: $\left(\frac{x^2 y^3}{z^4}\right)^2 = \frac{x^{4} y^{6}}{z^{8}}$

Zero and Negative Exponents

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

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

• 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 Standard and Scientific Notation

• To convert to scientific notation, move the decimal point so that only one nonzero digit remains to the left. Count the number of places moved to determine the exponent.

• To convert to standard form, move the decimal point to the right (for positive exponents) or left (for negative exponents) as indicated by the exponent.

• 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: $(7.6 \times 10^4) \times (2.3 \times 10^3) = (7.6 \times 2.3) \times 10^{4+3} = 17.48 \times 10^7 = 1.748 \times 10^8$

Polynomials

Definition and Structure

A polynomial is an algebraic expression consisting of terms in the form $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$, where the exponents are non-negative integers and the coefficients are real numbers.

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

• Coefficient: The numerical factor of a term

• Degree: The highest exponent of the variable

• Constant: A term with no variable

• Example: In $5x^3 - 2x^2 + 7x - 4$, the degree is 3, coefficients are 5, -2, 7, and the constant is -4.

Classifying Polynomials

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

Operations with Polynomials

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

• Multiplication: Use the distributive property or FOIL for binomials.

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

• Example: $(3x^2 + 2x) + (4x^2 - x) = 7x^2 + x$

Applications: Area and Scientific Context

Area Formulas

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

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

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

Scientific Constants and Notation

• Avogadro's Number: The number of atoms or molecules in one mole of a substance. $6.022 \times 10^{23}$

• Example: Write Avogadro's number in scientific notation: $602,214,199,000,000,000,000,000 = 6.02214199 \times 10^{23}$

Simple Interest and 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 interest is compounded per year

  • $t$ = number of years

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

Geometry and Algebraic Formulas

Solving for Variables in Formulas

• Isolate the desired variable using algebraic operations.

• Example: For $z = 2x^2 - \frac{7}{2}y$, solve for $y$:

$z = 2x^2 - \frac{7}{2}y$ $z - 2x^2 = -\frac{7}{2}y$ $y = -\frac{2}{7}(z - 2x^2)$

Summary Table: Key Exponent Rules

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

Additional info: This guide covers foundational College Algebra topics including exponents, polynomials, scientific notation, and applications in geometry and finance, as reflected in the provided practice problems and solutions.