Which of the following is equivalent to the expression (i⋅5–√)⋅3, which shows that the Associative Property of Multiplication holds true for complex numbers?(1 point)%0D%0AResponses%0D%0A%0D%0A15i%0D%0A15 i%0D%0A%0D%0A3i5–√%0D%0A3 i start square root 5 end square root%0D%0A%0D%0Ai15−−√%0D%0Ai start square root 15 end square root%0D%0A%0D%0Ai⋅(5–√⋅3)

1 answer

To simplify the expression \((i \cdot \sqrt{5}) \cdot 3\) and verify the Associative Property of Multiplication for complex numbers, we can rearrange the expression.

According to the Associative Property, we should be able to group the numbers in different ways without changing the product:

  1. Original expression: \((i \cdot \sqrt{5}) \cdot 3\)
  2. By re-grouping: \(i \cdot (\sqrt{5} \cdot 3)\)

Now, we can compute \(\sqrt{5} \cdot 3\):

\[\sqrt{5} \cdot 3 = 3\sqrt{5}\]

Therefore, we can express the original product as:

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

Now, let’s compare this to the options provided:

  1. 15i (which does not match)
  2. 3i√5 (this matches our result)
  3. i15−−√ (which does not match)
  4. i⋅(5−−√⋅3) (this is equivalent to the original expression but not simplified)

Thus, the equivalent expression for \((i \cdot \sqrt{5}) \cdot 3\) is \(3i\sqrt{5}\).

The answer is 3i√5.