College Algebra Practice Problems and Solutions: Exponents, Polynomials, and Scientific Notation

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: $a^m imes a^n = a^{m+n}$

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

Quotient Rule for Exponents

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

• Definition: $a^m ig/ a^n = a^{m-n}$

• Example: $x^7 ig/ x^4 = x^{7-4} = x^3$

Power Rule for Exponents

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

• Definition: $(a^m)^n = a^{mn}$

• Example: $(x^2)^3 = x^{2 imes 3} = x^6$

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: $(ab)^n = a^n b^n$

• Power of a Quotient: $(a/b)^n = a^n / b^n$

• Example: $(2x)^3 = 2^3 x^3 = 8x^3$

Zero and Negative Exponents

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

• Zero Exponent: $a^0 = 1$ (for $a eq 0$)

• Negative Exponent: $a^{-n} = 1/a^n$

• Example: $x^{-2} = 1/x^2$

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: $a imes 10^n$ where $1 ext{ ≤ } a < 10$ and $n$ is an integer

• Example: $5,870,000 = 5.87 imes 10^6$

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: $0.0000078 = 7.8 imes 10^{-6}$

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: $(2.7 imes 10^4) imes (3.2 imes 10^3) = (2.7 imes 3.2) imes 10^{4+3} = 8.64 imes 10^7$

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: $5x^2 - 3x + 7$ 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: $(x+2)(x+3) = x^2 + 5x + 6$

Polynomial Table Example

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

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

Geometry Applications in Algebra

Area Formulas

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

• Square: $A = ext{side}^2$

• Rectangle: $A = ext{length} imes ext{width}$

• Triangle: $A = rac{1}{2} imes ext{base} imes ext{height}$

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, $A = rac{1}{2} imes 2.4 imes 3.6 = 4.32$ 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: $A = P igg(1 + rac{r}{n}igg)^{nt}$

• 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: $P = 5000$, $r = 0.04$, $n = 12$, $t = 5$ $A = 5000 igg(1 + rac{0.04}{12}igg)^{12 imes 5}$

Additional Info

• Avogadro's number is a constant used in chemistry, but its representation in scientific notation is a useful algebraic skill: $6.022 imes 10^{23}$

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