Podcast
Questions and Answers
What is the additive inverse to a?
What is the additive inverse to a?
- expression^1/n
- 1/a
- 1/(expression)^n
- -a (correct)
What is the multiplicative inverse to a (when a does not equal 0)?
What is the multiplicative inverse to a (when a does not equal 0)?
1/a
What does raising an expression to a negative exponent equal?
What does raising an expression to a negative exponent equal?
1/b^n
How do you multiply expressions with powers?
How do you multiply expressions with powers?
What is the result of raising expressions to powers?
What is the result of raising expressions to powers?
What is the formula for dividing expressions with powers?
What is the formula for dividing expressions with powers?
What do fractional exponents represent?
What do fractional exponents represent?
What is the result of multiplying radicals?
What is the result of multiplying radicals?
What is the formula for factoring the difference of two squares?
What is the formula for factoring the difference of two squares?
What is the formula for factoring perfect squares?
What is the formula for factoring perfect squares?
How is the slope of a line determined given two points?
How is the slope of a line determined given two points?
What is the equation of a line in slope-intercept form?
What is the equation of a line in slope-intercept form?
What is the point-slope form of a line given two points?
What is the point-slope form of a line given two points?
What is the intercept form of a line given x and y intercepts?
What is the intercept form of a line given x and y intercepts?
What is the condition for slopes of perpendicular lines?
What is the condition for slopes of perpendicular lines?
How is the discriminant used to determine the number of real solutions?
How is the discriminant used to determine the number of real solutions?
What does logarithm addition state?
What does logarithm addition state?
What does logarithm subtraction state?
What does logarithm subtraction state?
What is the logarithm power rule?
What is the logarithm power rule?
What is the formula for an arithmetic sequence?
What is the formula for an arithmetic sequence?
What is the sum of arithmetic sequences?
What is the sum of arithmetic sequences?
What is the formula for a geometric sequence?
What is the formula for a geometric sequence?
What is the sum of geometric sequences?
What is the sum of geometric sequences?
What is the sum of infinite geometric sequences?
What is the sum of infinite geometric sequences?
What is the definition of a factorial?
What is the definition of a factorial?
What is the combination formula of n choose r?
What is the combination formula of n choose r?
What is the determinant of a 2x2 matrix represented as?
What is the determinant of a 2x2 matrix represented as?
How do you find the additive inverse of an expression?
How do you find the additive inverse of an expression?
What is the result of raising an expression to a negative number?
What is the result of raising an expression to a negative number?
What does raising to a fractional power signify?
What does raising to a fractional power signify?
How can you determine whether a graph is a function?
How can you determine whether a graph is a function?
How do you verify if a graph matches that of an algebraic function?
How do you verify if a graph matches that of an algebraic function?
How do you find the domain given y=f(x)?
How do you find the domain given y=f(x)?
How do you graph a line given y=mx+b?
How do you graph a line given y=mx+b?
What indicates whether a graph is odd, even, or neither?
What indicates whether a graph is odd, even, or neither?
How do you describe transformations given f(x)?
How do you describe transformations given f(x)?
How do you determine how many roots a given graph has?
How do you determine how many roots a given graph has?
What is the maximum and minimum number of roots for a polynomial of degree n?
What is the maximum and minimum number of roots for a polynomial of degree n?
How do you determine the roots given f(x)?
How do you determine the roots given f(x)?
Flashcards
Additive Inverse
Additive Inverse
The opposite of a number, which when added to the original number, results in zero.
Multiplicative Inverse
Multiplicative Inverse
The reciprocal of a non-zero number, which when multiplied by the original number, equals 1.
Negative Exponent
Negative Exponent
For b⁻ⁿ, it's 1/bⁿ.
Exponent Addition
Exponent Addition
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Exponent Raising
Exponent Raising
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Exponent Subtraction
Exponent Subtraction
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Fractional Exponents
Fractional Exponents
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Multiplying Radicals
Multiplying Radicals
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Difference of Squares
Difference of Squares
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Perfect Squares
Perfect Squares
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Cubes Factoring
Cubes Factoring
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Slope from Points
Slope from Points
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Slope-Intercept Form
Slope-Intercept Form
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Point-Slope Form
Point-Slope Form
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Intercept Form
Intercept Form
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Perpendicular Lines
Perpendicular Lines
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Discriminant
Discriminant
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Logarithm Addition
Logarithm Addition
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Logarithm Subtraction
Logarithm Subtraction
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Power Rule (Logs)
Power Rule (Logs)
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Arithmetic Sequence
Arithmetic Sequence
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Sum of Arithmetic Sequence
Sum of Arithmetic Sequence
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Geometric Sequence
Geometric Sequence
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Sum of Geometric Sequence
Sum of Geometric Sequence
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Infinite Geometric Series
Infinite Geometric Series
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Factorial
Factorial
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Combination
Combination
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Study Notes
Algebraic Concepts
- Additive Inverse: The additive inverse of a number 'a' is -a.
- Multiplicative Inverse: The multiplicative inverse of a non-zero number 'a' is 1/a.
- Negative Exponent Rule: Raising an expression 'b' to a negative exponent 'n' is equivalent to 1/b^n.
- Exponent Addition: When multiplying expressions with the same base, x^n * x^m = x^(m+n).
- Exponent Raising: (x^m)^n simplifies to x^(m*n).
- Exponent Subtraction: For division, x^m / x^n = x^(m-n).
Advanced Exponents and Radicals
- Fractional Exponents: x^(1/2) represents the square root of x, and x^(m/n) denotes the n-th root of x^m.
- Multiplying Radicals: The product of two square roots is √a * √b = √(ab).
Factoring Techniques
- Difference of Squares: x^2 - y^2 factors into (x - y)(x + y).
- Perfect Squares: x^2 + 2xy + y^2 = (x + y)^2 and x^2 - 2xy + y^2 = (x - y)^2.
- Cubes Factoring: x^3 + y^3 = (x + y)(x^2 - xy + y^2) and x^3 - y^3 = (x - y)(x^2 + xy + y^2).
Slope and Line Equations
- Slope from Points: Given two points, the slope (m) is calculated as m = (y2 - y1) / (x2 - x1).
- Slope-Intercept Form: The equation of a line can be expressed as y = mx + b, where m is the slope and b is the y-intercept.
- Point-Slope Form: For a line through point (x1, y1) with slope m: (y - y1) = m(x - x1).
- Intercept Form: Expressed as x/a + y/b = 1, where a and b are x and y intercepts respectively.
Graphing and Properties
- Perpendicular Line Slopes: Slopes of perpendicular lines are negative reciprocals of each other.
- Discriminant: Used to determine the number of real solutions:
- If b^2 - 4ac > 0, there are 2 real solutions.
- If b^2 - 4ac = 0, there is 1 real solution.
- If b^2 - 4ac < 0, there are no real solutions but 2 imaginary solutions.
Logarithmic Operations
- Addition of Logarithms: log a + log b = log(ab).
- Subtraction of Logarithms: log a - log b = log(a/b).
- Power Rule: log a^b = b log a.
Sequences
- Arithmetic Sequence: Defined as an_n = a1 + (n - 1)d, where d is the common difference.
- Sum of Arithmetic Sequences: Calculated using Sn = n/2[2a1 + (n - 1)d] or Sn = n/2(a1 + a2).
- Geometric Sequence: An expression a_n = a1 * r^(n-1), where r is the common ratio.
- Sum of Geometric Sequences: Calculated as Sn = [a1(1 - r^n)] / (1 - r).
- Infinite Geometric Series: Sn = a1 / (1 - r), valid for |r| < 1.
Combinatorics and Linear Algebra
- Factorial: Defined as n! = n(n - 1)(n - 2)...321.
- Combination Formula: nCr = n! / (r!(n - r)!).
- Determinant of a 2x2 Matrix: For a matrix with elements a, b, c, d, the determinant is ad - bc.
Additional Concepts
- Vertical Line Test: Used to determine whether a graph represents a function.
- Domain Determination: Domain exceptions occur when:
- The denominator equals zero.
- An even root yields a negative number.
- The logarithm of zero or a negative number occurs.
- Graph Symmetry: Functions are even if symmetrical about the y-axis and odd if symmetrical about the origin.
- Transformations: Adjustments to functions include vertical shifts, horizontal shifts, and reflections across axes.
Polynomial Roots
- Root Analysis: The number of roots corresponds to how many times the graph touches or crosses the x-axis.
- Max and Min Roots:
- For a polynomial of degree n, the maximum number of real roots is n.
- If n is even, the minimum is 0; if n is odd, the minimum is 1.
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Description
These flashcards are designed to help you master key concepts in college algebra, including inverses, exponents, and power properties. Each card provides a term along with its definition, making it easier for you to test your knowledge and prepare for the exam.