Question

Question 1(Multiple Choice Worth 2 points)
(01.06 MC)

Simplify i31.

1
−1
i
−i
Question 2(Multiple Choice Worth 2 points)
(01.06 LC)

Simplify square root of negative 121.

−11i
11i
−11
11
Question 3(Multiple Choice Worth 2 points)
(01.06 MC)

Simplify i38.

i
−1
−i
1
Question 4(Multiple Choice Worth 2 points)
(01.06 MC)

Simplify square root of negative 72.

negative 6 square root of 2
6 square root of negative 2
6 square root of 2 i
6 i square root of 2
Question 5(Multiple Choice Worth 2 points)
(01.06 MC)

Simplify negative 5 minus the square root of negative 44

negative 5 minus 4 times the square root of 11 i
negative 5 minus 4 i times the square root of 11
negative 5 minus 2 i times the square root of 11
negative 5 minus 2 times the square root of 11 i

Answer 1

Let's simplify each of the questions one by one:

**Question 1: Simplify i^31.**

The powers of \( i \) cycle through four values:
- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \)

To find \( i^{31} \), we can reduce the exponent modulo 4:
\[ 31 \mod 4 = 3 \]
Thus, \( i^{31} = i^3 = -i \).

**Answer: −i**

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**Question 2: Simplify \( \sqrt{-121} \).**

We can rewrite it as:
\[ \sqrt{-121} = \sqrt{121} \cdot \sqrt{-1} = 11i \].

**Answer: 11i**

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**Question 3: Simplify i^38.**

Again, using the cycle of powers of \( i \):
- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \)

Reduce 38 modulo 4:
\[ 38 \mod 4 = 2 \]
Thus, \( i^{38} = i^2 = -1 \).

**Answer: −1**

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**Question 4: Simplify \( \sqrt{-72} \).**

We can rewrite it as:
\[ \sqrt{-72} = \sqrt{72} \cdot \sqrt{-1} \]
\[ \sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2} \]
Thus:
\[ \sqrt{-72} = 6\sqrt{2}i \].

**Answer: 6 i square root of 2**

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**Question 5: Simplify \( -5 - \sqrt{-44} \).**

We can rewrite it as:
\[ -5 - \sqrt{-44} = -5 - \sqrt{44} \cdot \sqrt{-1} \]
\[ \sqrt{44} = \sqrt{4 \cdot 11} = 2\sqrt{11} \]
Thus:
\[ -\sqrt{-44} = -2\sqrt{11}i \]
So now we get:
\[ -5 - \sqrt{-44} = -5 - 2\sqrt{11}i \].

**Answer: negative 5 minus 2 i times the square root of 11**

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In summary, here are the answers to all questions:
1. −i
2. 11i
3. −1
4. 6 i square root of 2
5. negative 5 minus 2 i times the square root of 11