Overview
This lesson explains how to apply the product laws of exponents to algebraic expressions, including multiplying numbers and combining like bases using exponent rules.
Example 1: Multiplying Monomials with Exponents
- Multiply the coefficients first (e.g., -4 × 5 = -20).
- For like bases, add the exponents: (A^2 × A = A^{2+1} = A^3), (B × B^3 = B^{1+3} = B^4).
- Final answer: (-20A^3B^4), with the number first, then variables in alphabetical order.
Example 2: Power to a Power (Distributive Law)
- Apply the outside exponent to all terms inside parentheses: ((-5mn^3p^5)^2).
- ((-5)^2 = 25).
- (m^1) squared: (m^{1×2} = m^2).
- (n^3) squared: (n^{3×2} = n^6).
- (p^5) squared: (p^{5×2} = p^{10}).
- Final answer: (25m^2n^6p^{10}).
Example 3: Combining Product and Power Rules
- Distribute exponents inside each set of parentheses.
- For ((-2y^3z)^5): ((-2)^5 = -32), (y^{3×5} = y^{15}), (z^{1×5} = z^5).
- For ((6y^{10}z^8)^2): (6^2 = 36), (y^{10×2} = y^{20}), (z^{8×2} = z^{16}).
- Multiply outside results together: (-32 × 36 = -1,152), (y^{15} × y^{20} = y^{35}), (z^5 × z^{16} = z^{21}).
- Final answer: (-1,152y^{35}z^{21}).
Key Terms & Definitions
- Exponent — A small number written above and to the right of a base number, showing how many times the base is multiplied by itself.
- Product of Powers Rule — When multiplying like bases, add their exponents: (a^m × a^n = a^{m+n}).
- Power of a Power Rule — When raising a power to another power, multiply the exponents: ((a^m)^n = a^{mn}).
- Coefficient — The numerical factor in a term.
Action Items / Next Steps
- Practice multiplying expressions using product and power laws of exponents.
- Complete any assigned homework problems related to this lesson.