Which of the following is equivalent to the expression (i⋅5−)√⋅3

Question

(
𝑖
⋅
5
−
)
⋅
3
, which shows that the Associative Property of Multiplication holds true for complex numbers?(1 point)
Responses

3i5–√
3
𝑖
5
3 i square root of 5

i⋅(5–√⋅3)
𝑖
⋅
(
5
⋅
3
)
i times open paren square root of 5 times 3 close paren

i15−−√
𝑖
15
i square root of 15

15i
15
𝑖

Answer 1

To understand the equivalence in the expression \((i \cdot \sqrt{5}) \cdot 3\) and to show that the Associative Property of Multiplication holds true for complex numbers, let's break this down.

The Associative Property of Multiplication states that the way in which the factors are grouped does not affect the product. This property can be expressed mathematically as:

\((a \cdot b) \cdot c = a \cdot (b \cdot c)\)

For the given expression, we can analyze:

\[ (i \cdot \sqrt{5}) \cdot 3 \]

This can be regrouped as:

\[ i \cdot (\sqrt{5} \cdot 3) \]

We need to express this in terms of the possible responses given:

1. \( 3i\sqrt{5} \): This is equivalent to \( i \cdot (\sqrt{5} \cdot 3) \), since multiplication is commutative.

2. \( i \cdot (5\sqrt{3}) \): This is incorrect because it does not match the original expression.

3. \( i \sqrt{15} \): This results from \( \sqrt{5} \cdot 3 \), which is not specifically related to the structure of the original expression.

4. \( 15i \): This implies \( 15 \cdot i \), which does not represent the original expression.

The correct equivalent expression that demonstrates the Associative Property of Multiplication and matches the structure of \((i \cdot \sqrt{5}) \cdot 3\) is:

\( 3i\sqrt{5} \).