Simplify the following expression. square root-8 square root -24
sqrt -8 = i sqrt 8 = 2 sqrt 2 i
sqrt -24 = i sqrt 24 = i sqrt(8*3) = 2i sqrt2 sqrt3
so
2 i sqrt2 * 2 i sqrt2 sqrt3
=
-4*2 sqrt 3
=
-8 sqrt 3
To simplify the expression √(-8) √(-24), we can first simplify each square root separately, then multiply the results.
Let's start with √(-8). This involves finding the square root of a negative number, which is not a real number. However, we can rewrite it as a product involving the imaginary unit, i:
√(-8) = √((-1) * 8) = √(-1) * √(8) = i√8
Next, let's simplify √(-24) using the same process:
√(-24) = √((-1) * 24) = √(-1) * √(24) = i√24
Now we have i√8 and i√24. To simplify further, we can find the largest perfect square that divides into both 8 and 24. In this case, 8 is a perfect square, so we can simplify i√8 as 2i√2.
Substituting this simplification back into i√24, we have:
i√24 = i * √(4 * 6) = i * √4 * √6 = i * 2 * √6 = 2i√6.
Therefore, the simplified expression is:
√(-8) √(-24) = 2i√2 * 2i√6 = 4i^2√12.
Since i^2 = -1, the expression simplifies further to:
4i^2√12 = 4(-1)√12 = -4√12.
So, the simplified expression is -4√12.