For homework, Crystal must simplify the expression $\sqrt { - 3 } \cdot \sqrt { - 3 }$. She gets the answer $3$, but when she checks, she learns that the correct answer is $–3$.

1. Show Crystal the steps she could take to get $–3$.

   Step 1(a):

   $\sqrt{-3} \cdot \sqrt{-3} = i\sqrt{3} \cdot i\sqrt{3}$

   Step 2(a):

   $i^2\sqrt{9}$

   Step 3(a):

   $-1\sqrt{9}$

   Step 4(a):

   $-1 \cdot 3 = -3$

2. What steps do you think Crystal took to get $3$ as a result?

   Hint (b):

   What is $(−3)(−3)$?

3. What does she need to consider to avoid making this mistake in the future?

   Answer (c):

   She needs to convert to imaginary form before completing operations.

4. Crystal’s example means that $\sqrt { a } \cdot \sqrt { b } = \sqrt { a b }$ is not always true for real numbers $a$ and $b$. What restriction needs to be placed on the numbers $a$ and $b$?

   Answer (d):

   They must be non-negative real numbers.