Questions and Answers
What is the base in an exponential expression?
What does the exponent in an exponential expression indicate?
What is a monomial?
A one-term polynomial
What is a polynomial?
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What is the result of the product -a^2b^2c^2(a + b - c)?
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What do you get when you simplify 8y(y - 4) + 3y(y + 2)?
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What is the result of simplifying p(p - q) - q(q - p)?
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What do you get when you simplify 8 + 3{x - 2[x + 5(x + 3)]}?
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What is the product of 3a^2 and 2a^3?
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What do you get when you simplify 7(n - p) + 5p?
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What is the product of -2xa(-4xb - 2x^3 + 5x)?
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What is the product of 2y^2(3x + 5z)?
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What do you get when you find the product of a^4(3a^2 - 2a + 1)?
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What is the product of 3(m + n)?
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What do you get from -1/2y(2y^3 - 8)?
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What is the product of 4xy and y^2 + 2x?
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What is the product of -7(8 + k)?
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What do you get when you find the product of -4x^3y^2(7xy^4)?
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What do you get from simplifying 1/4(8m - 4n) + 1/3(6m + 3n)?
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What is the product of x^3(x^2 + 5x + 1)?
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What is the method for multiplying monomials?
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Study Notes
Key Definitions
- Base: The repeated factor in an exponential expression.
- Exponent: The superscript number indicating how many times the base is multiplied by itself.
- Monomial: A polynomial consisting of a single term.
- Polynomial: A mathematical expression composed of one or more terms with whole number exponents.
Examples of Products
- Product Calculation: For (-a^2b^2c^2(a + b - c)), the result is (-a^3b^2c^2 - a^2b^3c^2 + a^2b^2c^3).
- Simplification Example: (8y(y - 4) + 3y(y + 2)) simplifies to (11y^2 - 26y).
Algebraic Expressions Simplified
- For (p(p - q) - q(q - p)), the simplified form is (p^2 - q^2).
- Simplifying (8 + 3{x - 2[x + 5(x + 3)]}) leads to (-33x - 82).
Product Finding
- For (3a^2 · 2a^3), the product is (6a^5).
- The product of (7(n - p) + 5p) results in (7n - 2p).
Multiplying with Variables
- Multiplying (-2xa(-4xb - 2x^3 + 5x)) yields (8x^{a+b} + 4x^{a+3} - 10x^{a+1}).
- The expression (2y^2(3x + 5z)) results in (6xy^2 + 10y^2z).
Additional Product Examples
- For (a^4(3a^2 - 2a + 1)), the product is (3a^6 - 2a^5 + 1a^4).
- The product (3(m + n)) simplifies to (3m + 3n).
- The expression (-\frac{1}{2}y(2y^3 - 8)) simplifies to (-y^4 + 4y).
Summarizing Multiplication
- The product of (4xy) and (y^2 + 2x) results in (4xy^3 + 8x^2y).
- Simplifying (-7(8 + k)) gives (-7k - 56).
- For (-4x^3y^2(7xy^4)), the simplified product is (-28x^4y^6).
Final Simplifications
- The expression (\frac{1}{4}(8m - 4n) + \frac{1}{3}(6m + 3n)) simplifies to (4m).
- Finding the product (x^3(x^2 + 5x + 1)) results in (x^5 + 5x^4 + x^3).
Review of Key Concepts
- To multiply monomials, multiply coefficients and like bases.
- Distribute monomials across polynomials with multiple terms.
- Simplification involves removing grouping symbols through distribution and combining like terms.
Studying That Suits You
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Description
Test your knowledge on important algebraic terms with these flashcards focused on multiplying by a monomial. Learn the definitions of key concepts like base, exponent, monomial, and polynomial. Perfect for quick reviews and enhancing your understanding of algebra basics.
