Answer:
[tex]\begin{aligned}ab(a+b) & = (ab)a+(ab)b & & \textsf{Distributive Property of Addition} \\& = a(ab)+(ab)b & & \textsf{Commutative Property of Multiplication} \\& = (a \cdot a)b + a(b \cdot b) & & \textsf{Associative Property of Multiplication}\\& = a^2b+ab^2 & & \textsf{Property of Exponents}\end{aligned}[/tex]
Step-by-step explanation:
Distributive Property of Addition
Multiplying a number by a group of numbers added together is the same as multiplying each number separately.
Addition: a(b + c) = ab + ac
Subtraction: a(b - c) = ab – ac
Commutative Property
Changing the order or position of two numbers does not change the end result.
Applies to addition and multiplication only.
Addition: a + b = b + a
Multiplication: a × b = b × a
Associative Property
Grouping of numbers by parentheses in a different way does not affect their sum or product.
Applies to addition and multiplication only.
Addition: (a + b) + c = a + (b + c) = (a + c) + b
Multiplication: (a × b) × c = a × (b × c) = (a × c) × b
Property of Exponents
The exponent of a number shows how many times it should be multiplied by itself.
a × a × a = a³
b × b × b × b = b⁴
[tex]\begin{aligned}ab(a+b) & = (ab)a+(ab)b & & \textsf{Distributive Property of Addition} \\& = a(ab)+(ab)b & & \textsf{Commutative Property of Multiplication} \\& = (a \cdot a)b + a(b \cdot b) & & \textsf{Associative Property of Multiplication}\\& = a^2b+ab^2 & & \textsf{Property of Exponents}\end{aligned}[/tex]
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