Question3. Sinplify $\sqrt{-24}$

Studdy Solution

STEP 1

1. The expression involves a square root of a negative number, which means it will result in an imaginary number.
2. We will simplify the expression using imaginary numbers.

STEP 2

1. Express the negative number under the square root as a product of a positive number and $-1$ .
2. Simplify the square root by separating the product into two separate square roots.
3. Simplify the square root of the positive number and the square root of $-1$ .

STEP 3

Express $-24$ as a product of a positive number and $-1$ :
$\sqrt{-24} = \sqrt{24 \times (-1)}$

STEP 4

Separate the square root of the product into two separate square roots:
$\sqrt{24 \times (-1)} = \sqrt{24} \times \sqrt{-1}$

STEP 5

Simplify $\sqrt{24}$ by finding the prime factorization of 24:
$24 = 2^3 \times 3$
Thus,
$\sqrt{24} = \sqrt{2^3 \times 3} = \sqrt{2^2 \times 2 \times 3} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6}$

STEP 6

Simplify $\sqrt{-1}$ as $i$ , the imaginary unit:
$\sqrt{-1} = i$

STEP 7

Combine the results from the previous steps:
$\sqrt{-24} = 2\sqrt{6} \times i = 2i\sqrt{6}$
The simplified form of $\sqrt{-24}$ is:
$\boxed{2i\sqrt{6}}$

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