Answer :
Let's examine each expression to see whether we need to use the distributive property for simplification.
1. Expression [tex]\( \sqrt{5}(-\sqrt{2}) \)[/tex]:
Here, we are multiplying two terms: [tex]\( \sqrt{5} \)[/tex] and [tex]\( -\sqrt{2} \)[/tex]. Since there are no additions or subtractions within a parenthesis involved, we can directly multiply the two terms. Thus, the distributive property is not required.
2. Expression [tex]\( \sqrt{5}(\sqrt{7} - \sqrt{2}) \)[/tex]:
In this expression, we are multiplying [tex]\( \sqrt{5} \)[/tex] with the binomial [tex]\( (\sqrt{7} - \sqrt{2}) \)[/tex]. To simplify, we need to distribute [tex]\( \sqrt{5} \)[/tex] to each term inside the parentheses:
[tex]\[ \sqrt{5}(\sqrt{7} - \sqrt{2}) = \sqrt{5} \cdot \sqrt{7} - \sqrt{5} \cdot \sqrt{2} \][/tex]
Hence, the distributive property is used in this case.
3. Expression [tex]\( (\sqrt{5} + \sqrt{2})(-\sqrt{7}) \)[/tex]:
Here, we have a binomial [tex]\( (\sqrt{5} + \sqrt{2}) \)[/tex] being multiplied by a single term [tex]\( -\sqrt{7} \)[/tex]. To simplify, we need to distribute [tex]\( -\sqrt{7} \)[/tex] to each term inside the parenthesis:
[tex]\[ (\sqrt{5} + \sqrt{2})(-\sqrt{7}) = \sqrt{5} \cdot (-\sqrt{7}) + \sqrt{2} \cdot (-\sqrt{7}) \][/tex]
Therefore, the distributive property is used in this instance as well.
4. Expression [tex]\( (3 \sqrt{5})(-7 \sqrt{2}) \)[/tex]:
In this expression, we are multiplying two terms: [tex]\( 3 \sqrt{5} \)[/tex] and [tex]\( -7 \sqrt{2} \)[/tex]. There are no additions or subtractions within a parenthesis involved, so we can directly multiply the two terms. The distributive property is not necessary here.
Based on this analysis, the expressions that require the use of the distributive property to simplify are:
[tex]\[ \sqrt{5}(\sqrt{7} - \sqrt{2}) \][/tex]
[tex]\[ (\sqrt{5} + \sqrt{2})(-\sqrt{7}) \][/tex]
Thus, the correct selections are:
[tex]\[ \boxed{2 \text{ and } 3} \][/tex]
1. Expression [tex]\( \sqrt{5}(-\sqrt{2}) \)[/tex]:
Here, we are multiplying two terms: [tex]\( \sqrt{5} \)[/tex] and [tex]\( -\sqrt{2} \)[/tex]. Since there are no additions or subtractions within a parenthesis involved, we can directly multiply the two terms. Thus, the distributive property is not required.
2. Expression [tex]\( \sqrt{5}(\sqrt{7} - \sqrt{2}) \)[/tex]:
In this expression, we are multiplying [tex]\( \sqrt{5} \)[/tex] with the binomial [tex]\( (\sqrt{7} - \sqrt{2}) \)[/tex]. To simplify, we need to distribute [tex]\( \sqrt{5} \)[/tex] to each term inside the parentheses:
[tex]\[ \sqrt{5}(\sqrt{7} - \sqrt{2}) = \sqrt{5} \cdot \sqrt{7} - \sqrt{5} \cdot \sqrt{2} \][/tex]
Hence, the distributive property is used in this case.
3. Expression [tex]\( (\sqrt{5} + \sqrt{2})(-\sqrt{7}) \)[/tex]:
Here, we have a binomial [tex]\( (\sqrt{5} + \sqrt{2}) \)[/tex] being multiplied by a single term [tex]\( -\sqrt{7} \)[/tex]. To simplify, we need to distribute [tex]\( -\sqrt{7} \)[/tex] to each term inside the parenthesis:
[tex]\[ (\sqrt{5} + \sqrt{2})(-\sqrt{7}) = \sqrt{5} \cdot (-\sqrt{7}) + \sqrt{2} \cdot (-\sqrt{7}) \][/tex]
Therefore, the distributive property is used in this instance as well.
4. Expression [tex]\( (3 \sqrt{5})(-7 \sqrt{2}) \)[/tex]:
In this expression, we are multiplying two terms: [tex]\( 3 \sqrt{5} \)[/tex] and [tex]\( -7 \sqrt{2} \)[/tex]. There are no additions or subtractions within a parenthesis involved, so we can directly multiply the two terms. The distributive property is not necessary here.
Based on this analysis, the expressions that require the use of the distributive property to simplify are:
[tex]\[ \sqrt{5}(\sqrt{7} - \sqrt{2}) \][/tex]
[tex]\[ (\sqrt{5} + \sqrt{2})(-\sqrt{7}) \][/tex]
Thus, the correct selections are:
[tex]\[ \boxed{2 \text{ and } 3} \][/tex]
