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

Question

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

15i
15 i

3i5–√
3 i square root of 5

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

i15−−√

Answer 1

To determine which expression is equivalent to \((i \cdot \sqrt{5}) \cdot \sqrt{3}\) and shows the Associative Property of Multiplication holds true for complex numbers, we can apply the associative property, which states that the way in which numbers are grouped in multiplication does not change the product.

We start with the given expression:

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

By the associative property, we can regroup this expression as:

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

Since \(\sqrt{a}\cdot \sqrt{b} = \sqrt{a \cdot b}\), we can simplify \(\sqrt{5} \cdot \sqrt{3}\) to \(\sqrt{15}\):

\[ i \cdot \sqrt{15} \]

Now looking at the options:

1. 15i - Not equivalent.
2. 3i \sqrt{5} - Not equivalent.
3. i \cdot (\sqrt{5} \cdot \sqrt{3}) - This is equivalent to \(i \cdot \sqrt{15}\).
4. i \sqrt{15} - This is equivalent to \(i \cdot \sqrt{15}\) as derived.

The correct response that is equivalent to the expression and shows the Associative Property of Multiplication is:

i⋅(5−−√⋅3) (option 3).