Podcast
Questions and Answers
Which law of exponents applies when multiplying two powers with the same base?
Which law of exponents applies when multiplying two powers with the same base?
- Product of Powers Property (correct)
- Power of a Power Property
- Quotient of Powers Property
- Power of a Product Property
What is the solution to the linear equation $2x + 4 = 12$?
What is the solution to the linear equation $2x + 4 = 12$?
- 4
- 6 (correct)
- 2
- 8
Using the method of Factoring by Grouping, which expression can be factored as $x(x + 3) + 4(x + 3)$?
Using the method of Factoring by Grouping, which expression can be factored as $x(x + 3) + 4(x + 3)$?
- $x^2 + 3x + 12$
- $x^2 + 4x + 12$
- $x^2 + 7x + 12$ (correct)
- $x^2 + 3x + 4$
Which factoring method would be best used for the expression $x^2 - 9$?
Which factoring method would be best used for the expression $x^2 - 9$?
Which method is used to factor the expression $x^3 + 27$?
Which method is used to factor the expression $x^3 + 27$?
What is the result of applying the law of exponents when raising a power to another power, specifically for the expression $(x^3)^4$?
What is the result of applying the law of exponents when raising a power to another power, specifically for the expression $(x^3)^4$?
Which factoring method is most suitable for factoring the expression $x^4 - 16$?
Which factoring method is most suitable for factoring the expression $x^4 - 16$?
What is the general form of a linear equation in one variable?
What is the general form of a linear equation in one variable?
When factoring the expression $2x^2 + 8x$ using the greatest common factor method, what will the factored form be?
When factoring the expression $2x^2 + 8x$ using the greatest common factor method, what will the factored form be?
Which of the following describes the factoring process of $x^3 - 3x^2 + 4x$ using the method of factoring by grouping?
Which of the following describes the factoring process of $x^3 - 3x^2 + 4x$ using the method of factoring by grouping?
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Study Notes
Laws of Exponents
- Product of Powers: When multiplying like bases, add the exponents (a^m * a^n = a^(m+n)).
- Quotient of Powers: When dividing like bases, subtract the exponents (a^m / a^n = a^(m-n)).
- Power of a Power: When raising a power to another power, multiply the exponents ((a^m)^n = a^(m*n)).
- Power of a Product: Distribute the exponent to each base in the product ((ab)^n = a^n * b^n).
- Power of a Quotient: Distribute the exponent to each base in the quotient ((a/b)^n = a^n / b^n).
- Zero Exponent: Any non-zero base raised to the zero power equals one (a^0 = 1).
- Negative Exponent: A negative exponent indicates a reciprocal (a^-n = 1/a^n).
Linear Equations in One Variable
- Form: Standard form for a linear equation is ax + b = 0, where a and b are constants.
- Solution: The solution is found by isolating the variable x (x = -b/a).
- Graph Interpretation: Represents a straight line on a coordinate plane.
- Properties: Linear equations can have one solution, infinitely many solutions, or no solution based on the relationship between coefficients.
Factoring Methods
- CMF (Common Monomial Factor): Factor out the greatest common factor from all terms (e.g., from ax + ay, factor out a to get a(x+y)).
- PST (Perfect Square Trinomial): Recognize and factor expressions of the form a^2 ± 2ab + b^2 (e.g., x^2 + 6x + 9 = (x+3)^2).
- DTPS (Difference of Two Perfect Squares): Factor expressions of the form a^2 - b^2 into (a-b)(a+b) (e.g., x^2 - 16 = (x-4)(x+4)).
- SDTPC (Sum and Difference of Two Perfect Cubes): Apply formulas to factor a^3 ± b^3:
- For sum: a^3 + b^3 = (a+b)(a^2 - ab + b^2)
- For difference: a^3 - b^3 = (a-b)(a^2 + ab + b^2).
- GT (Group Terms): Combine and factor using grouping; group terms in pairs and factor out common factors (e.g., from ax + ay + bx + by, factor to get (a+b)(x+y)).
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