5a²b⁵c³ . -4a⁷c⁵
To multiply these expressions, we can multiply the coefficients (numbers) separately and then multiply the variables together:
5a²b⁵c³ . -4a⁷c⁵ = (-4)(5)(a²)(a⁷)(b⁵)(c³)(c⁵)
Simplifying the coefficients: (-4)(5) = -20
Simplifying the variables:
a² * a⁷ = a^(2 + 7) = a⁹
b⁵ * c⁵ = b^(5) * c^(3+5) = b⁵c⁸
Putting it all together: -20a⁹b⁵c⁸
2x²y⁴z⁵ . 4x³y⁵z²
To simplify the expression 2x²y⁴z⁵ . 4x³y⁵z², we can multiply the coefficients separately and then multiply the variables together:
2x²y⁴z⁵ . 4x³y⁵z² = (2)(4)(x²)(x³)(y⁴)(y⁵)(z⁵)(z²)
Simplifying the coefficients: (2)(4) = 8
Simplifying the variables:
x² * x³ = x^(2 + 3) = x⁵
y⁴ * y⁵ = y^(4 + 5) = y⁹
z⁵ * z² = z^(5 + 2) = z⁷
Putting it all together: 8x⁵y⁹z⁷
3a³b² (6ab + 7a²b)
To simplify the expression, we need to use the distributive property to multiply each term inside the parentheses by 3a³b²:
3a³b² * 6ab = 18a^(3 + 1)b^(2 + 1) = 18a⁴b³
3a³b² * 7a²b = 21a^(3 + 2)b^(2 + 1) = 21a⁵b³
Combining the two terms:
18a⁴b³ + 21a⁵b³ = 39a⁵b³ + 18a⁴b³ = (39a⁵ + 18a⁴) * b³
The simplified expression is (39a⁵ + 18a⁴) * b³.