Question

How do you use the distributive property to simplify \[7\left( {y + 8} \right)\]?

Answer 1

Hint: Multiplication operation has a unique property called distribution over another operation addition or subtraction. It says that the product of a number to the sum of other two numbers is equal to the sum of the product of the first number to other numbers individually.

Complete step by step solution:
The given algebraic expression is shown below.
\[7\left( {y + 8} \right)\]
We generally omit the sign of multiplication before parenthesis. So, we can represent this algebraic expression as shown below.
\[ \Rightarrow 7 \times \left( {y + 8} \right)\]
Here, number seven is multiplying with parentheses that contain the sum of variable y and number eight.
It is known that multiplication operation has a unique property called distribution over another operation addition or subtraction. It says that product of a number \[a\] to the sum of other two numbers \[\left( {b + c} \right)\] is equal to the sum of the product of first number to other numbers individually \[a \times b + a \times c\].
Thus, we can resolve the expression by the use of distributive property of multiplication over addition as shown below.
We multiply number seven with variable y and add it to the product of number seven and number eight by the use of the distributive property of multiplication over addition.
Therefore, in mathematical form we can simplify the algebraic expression as shown below.\[ \Rightarrow 7 \times y + 7 \times 8\]
Simplify the expression by omitting the symbol of multiplication in the first term of expression as shown below.
\[ \Rightarrow 7y + 7 \times 8\]
Further simplify the expression as substituting the product of number seven and number eight in the second term.
\[ \Rightarrow 7y + 56\]
Thus, the simplified expression for algebraic expression \[7\left( {y + 8} \right)\] by the use of distributive property is \[7y + 56\].

Note:
Distributive and Commutative properties of algebraic operations are the basic rules to simplify any algebraic expression. Like for multiplication, \[a \times b = b \times a\] is a commutative property and \[a\left( {b + c} \right) = ab + ac\] is a distributive property.