Simplify the expression 5 sqrt(9) - sqrt(-64). Write your answer as a complex number.

Step-by-step solution

1. 1

   Separate the real radical from the imaginary one. $\sqrt{9}$ is an ordinary real square root, but $\sqrt{-64}$ has a negative radicand and has no real value at all. Treating the two the same way is what makes this problem a trap, so handle them one at a time.

2. 2

   Evaluate the real part. The principal square root of $9$ is $3$ (not $\pm 3$ — the radical symbol denotes the non-negative root), so

   $5\sqrt{9} = 5 \cdot 3 = 15$

3. 3

   Rewrite the negative radical with $i$. Pull out $-1$ using $\sqrt{-a} = i\sqrt{a}$ for $a>0$:

   $\sqrt{-64} = \sqrt{64 \cdot (-1)} = \sqrt{64}\,\sqrt{-1} = 8i$

   Writing $\sqrt{-64} = -8$ is the standard error: $(-8)^2 = +64$, not $-64$.

4. 4

   Subtract, keeping real and imaginary parts apart. Real terms combine with real terms and imaginary with imaginary; $15$ and $8i$ are unlike, so nothing merges:

   $15 - 8i$

5. 5

   Check the answer. In the form $a + bi$ we have $a = 15$ and $b = -8$. Squaring the imaginary part back: $(8i)^2 = 64i^2 = -64$ ✓, confirming $\sqrt{-64} = 8i$. The final value is therefore $15 - 8i$, and a common wrong option, $15 - 8$, would collapse to the real number $7$.