Question

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

Answer 1

To demonstrate the Associative Property of Multiplication for complex numbers, we need to show that the grouping of the numbers does not affect the product.

Let's consider the original expression \( (i \cdot \sqrt{5}) \cdot 3 \). The Associative Property states that:

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

So we can rewrite the expression as follows:

1. First, let's rewrite the original expression for clarity:
\[ (i \cdot \sqrt{5}) \cdot 3 \]

2. According to the Associative Property:
\[ i \cdot (\sqrt{5} \cdot 3) \]

Now, we can evaluate \( \sqrt{5} \cdot 3 \), which is:
\[ 3\sqrt{5} \]

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

This means that:
\[ (i \cdot \sqrt{5}) \cdot 3 = i \cdot (3 \cdot \sqrt{5}) \]

Now, let's analyze the given options:

1. **\(3i\sqrt{5}\)** is equivalent to \( (i \cdot \sqrt{5}) \cdot 3 \), but it does not represent the grouping we expressed.

2. **\(i \cdot (5 \cdot \sqrt{3})\)** clearly does not match with our expression.

3. **\(i\sqrt{15}\)** also does not fit.

4. **\(15i\)** does not represent our grouping either.

None of the responses exactly match the mathematical illustration of the Associative Property with the expression \( (i \cdot \sqrt{5}) \cdot 3 \).

Hence, the answer that will best support the associative property in this context is **\(i \cdot (3\sqrt{5})\)**, which is not listed in the options given. Please check with your question options, as they do not reflect the associative identity accurately.