Question

select all of the following expressions that require use of the distributive property to simplify.

\\(\sqrt{5}(-\sqrt{2})\\)

\\(\sqrt{5}(\sqrt{7} - \sqrt{2})\\)

\\((\sqrt{5} + \sqrt{2})(-\sqrt{7})\\)

\\((3\sqrt{5})(-7\sqrt{2})\\)

Explanation:

Define the distributive property

The distributive property states that multiplying a sum or difference by a number gives the same result as multiplying each addend individually by the number and then adding or subtracting the products:

$$a(b + c) = ab + ac$$

$$(a + b)c = ac + bc$$

We need to identify which expressions involve multiplying a single term by a sum or difference of terms.

Analyze the first expression

The first expression is:

$$\sqrt{5}(-\sqrt{2})$$

This is a product of two single terms (monomials). It does not involve a sum or difference inside parentheses. Thus, it only requires the associative/commutative properties of multiplication and radical multiplication rules, not the distributive property.

Analyze the second expression

The second expression is:

$$\sqrt{5}(\sqrt{7} - \sqrt{2})$$

Here, the term \(\sqrt{5}\) is multiplied by a binomial difference \(\sqrt{7} - \sqrt{2}\). To simplify, we must distribute \(\sqrt{5}\) to both terms inside the parentheses:

$$\sqrt{5}(\sqrt{7} - \sqrt{2}) = \sqrt{5}\cdot\sqrt{7} - \sqrt{5}\cdot\sqrt{2}$$

This requires the distributive property.

Analyze the third expression

The third expression is:

$$(\sqrt{5} + \sqrt{2})(-\sqrt{7})$$

Here, a binomial sum \(\sqrt{5} + \sqrt{2}\) is multiplied by a single term \(-\sqrt{7}\). To simplify, we must distribute \(-\sqrt{7}\) to both terms inside the parentheses:

$$(\sqrt{5} + \sqrt{2})(-\sqrt{7}) = \sqrt{5}(-\sqrt{7}) + \sqrt{2}(-\sqrt{7})$$

This requires the distributive property.

Analyze the fourth expression

The fourth expression is:

$$(3\sqrt{5})(-7\sqrt{2})$$

This is a product of two single terms (monomials). Simplifying this expression involves rearranging the factors using the commutative and associative properties of multiplication:

$$(3\sqrt{5})(-7\sqrt{2}) = (3 \cdot -7)(\sqrt{5} \cdot \sqrt{2})$$

This does not require the distributive property.

Answer:

• \(\sqrt{5}(-\sqrt{2})\)
• \(\sqrt{5}(\sqrt{7} - \sqrt{2})\) (Correct answer)
• \((\sqrt{5} + \sqrt{2})(-\sqrt{7})\) (Correct answer)
• \((3\sqrt{5})(-7\sqrt{2})\)