Math

Question Evaluate 36\sqrt{-36} and 81-\sqrt{81}. Click "Not a real number" if applicable.

Studdy Solution

STEP 1

Assumptions
1. The square root of a negative number is not a real number and is instead an imaginary number.
2. The square root of a positive number is a real number.
3. The negative square root of a positive number is also a real number.

STEP 2

Evaluate the first expression, 36\sqrt{-36}.

STEP 3

Since the square root of a negative number is not a real number, we express it using the imaginary unit ii, where i2=1i^2 = -1.

STEP 4

Factor out the negative sign from under the square root to show the imaginary unit.
36=36(1)\sqrt{-36} = \sqrt{36 \cdot (-1)}

STEP 5

Separate the square root of the positive number and the square root of 1-1.
36=361\sqrt{-36} = \sqrt{36} \cdot \sqrt{-1}

STEP 6

Calculate the square root of the positive number.
36=6\sqrt{36} = 6

STEP 7

Replace the square root of 1-1 with the imaginary unit ii.
1=i\sqrt{-1} = i

STEP 8

Combine the results to find the value of the first expression.
36=6i\sqrt{-36} = 6i

STEP 9

Now evaluate the second expression, 81-\sqrt{81}.

STEP 10

Calculate the square root of the positive number 81.
81=9\sqrt{81} = 9

STEP 11

Apply the negative sign in front of the square root.
81=9-\sqrt{81} = -9

STEP 12

Combine the results to provide the final answers for both expressions.
The value of 36\sqrt{-36} is 6i6i and the value of 81-\sqrt{81} is 9-9.

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