Question

Name the property used in each step.

Answer 1

Answer:

$\begin{aligned}ab(a+b) & = (ab)a+(ab)b & & \textsf{Distributive Property of Addition} \\& = a(ab)+(ab)b & & \textsf{Commutative Property of Multiplication} \\& = (a \cdot a)b + a(b \cdot b) & & \textsf{Associative Property of Multiplication}\\& = a^2b+ab^2 & & \textsf{Property of Exponents}\end{aligned}$

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

$\begin{aligned}ab(a+b) & = (ab)a+(ab)b & & \textsf{Distributive Property of Addition} \\& = a(ab)+(ab)b & & \textsf{Commutative Property of Multiplication} \\& = (a \cdot a)b + a(b \cdot b) & & \textsf{Associative Property of Multiplication}\\& = a^2b+ab^2 & & \textsf{Property of Exponents}\end{aligned}$

Learn more about Property Laws here:

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