Look at each simplified expression. Recognizing operations in an expression will guide you in how to simplify it.
[tex]
\begin{array}{l}
\sqrt{5}(-\sqrt{2})=-\sqrt{10} \\
\sqrt{5}(\sqrt{7}-\sqrt{2})-\sqrt{35}-\sqrt{10} \\
(\sqrt{5}+\sqrt{2})(-\sqrt{7})=-\sqrt{35}-\sqrt{14} \\
(3 \sqrt{5})(-7 \sqrt{2})=-21 \sqrt{10}
\end{array}
[/tex]
Choose two of the expressions that require the use of the distributive property to simplify.
[tex](\sqrt{5}+\sqrt{2})(-\sqrt{7})[/tex]
[tex](3 \sqrt{5})(-7 \sqrt{2})[/tex]
[tex]\sqrt{5}(-\sqrt{2})[/tex]
[tex]\sqrt{5}(\sqrt{7}-\sqrt{2})[/tex]
Answer (2)
The distributive property is used when multiplying a term by a sum or difference of terms.
Examine each expression to identify if it contains a term multiplied by a sum or difference.
5 ( 7 − 2 ) requires the distributive property: 5 × 7 − 5 × 2 = 35 − 10 .
( 5 + 2 ) ( − 7 ) requires the distributive property: − 7 × 5 − 7 × 2 = − 35 − 14 .
Explanation
Understanding the Distributive Property We need to identify which of the given expressions require the distributive property to simplify. The distributive property states that a ( b + c ) = ab + a c . This means we are looking for expressions where a term is multiplied by a sum or difference of terms.
Analyzing Each Expression Let's examine each expression:
5 ( − 2 ) : This expression involves only multiplication. There is no sum or difference of terms, so the distributive property is not required.
5 ( 7 − 2 ) : This expression involves a term, 5 , multiplied by a difference, ( 7 − 2 ) . Therefore, the distributive property is required to simplify this expression: 5 × 7 − 5 × 2 = 35 − 10 .
( 5 + 2 ) ( − 7 ) : This expression involves a sum, ( 5 + 2 ) , multiplied by a term, ( − 7 ) . Therefore, the distributive property is required to simplify this expression: − 7 × 5 − 7 × 2 = − 35 − 14 .
( 3 5 ) ( − 7 2 ) : This expression involves only multiplication. There is no sum or difference of terms, so the distributive property is not required.
Identifying the Expressions Based on our analysis, the two expressions that require the distributive property are 5 ( 7 − 2 ) and ( 5 + 2 ) ( − 7 ) .
Examples
The distributive property is a fundamental concept in algebra and is used in many real-world applications. For example, when calculating the total cost of purchasing multiple items with different prices, you use the distributive property. If you buy 3 apples at $0.50 e a c han d 2 banana s a t $0.75 each, the total cost can be calculated as 3($0.50) + 2($0.75), which is an application of the distributive property.
The two expressions that require the use of the distributive property are \sqrt\ 5 (\sqrt\ 7 -\sqrt\ 2 ) and ( \sqrt\ 5 +\sqrt\ 2 ) ( − \sqrt\ 7 ) . The other expressions do not involve any sums or differences requiring distribution.
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