Answer :
Let's go through each part of the question step-by-step.
### 1. Simplify without a calculator:
#### a) [tex]\((-3)^2\)[/tex]
This means [tex]\(-3\)[/tex] raised to the power of 2:
[tex]\[ (-3)^2 = 9 \][/tex]
#### b) [tex]\(-2^3\)[/tex]
This means the negative of [tex]\(2\)[/tex] raised to the power of 3:
[tex]\[ -2^3 = -(2 \times 2 \times 2) = -8 \][/tex]
#### d) [tex]\(\sqrt{-125}\)[/tex]
The square root of a negative number is not defined in real numbers. Hence:
[tex]\[ \sqrt{-125} = \text{undefined in real numbers} \][/tex]
(in complex numbers, it would be [tex]\(5i\sqrt{5}\)[/tex])
#### e) [tex]\(\sqrt{49}\)[/tex]
This is the square root of 49:
[tex]\[ \sqrt{49} = 7 \][/tex]
#### g) [tex]\(-3 \times \sqrt[3]{64}\)[/tex]
[tex]\[ \sqrt[3]{64} = 4 \][/tex]
So,
[tex]\[ -3 \times 4 = -12 \][/tex]
#### h) [tex]\(-2 \times \sqrt[3]{-27}\)[/tex]
[tex]\[ \sqrt[3]{-27} = -3 \][/tex]
So,
[tex]\[ -2 \times -3 = 6 \][/tex]
#### j) [tex]\(4^2-\sqrt[3]{64}\)[/tex]
[tex]\[ 4^2 = 16 \][/tex]
[tex]\[ \sqrt[3]{64} = 4 \][/tex]
So,
[tex]\[ 16 - 4 = 12 \][/tex]
#### k) [tex]\(((-1)^2) \times((-1)^3) \div(-1)\)[/tex]
[tex]\[ (-1)^2 = 1 \][/tex]
[tex]\[ (-1)^3 = -1 \][/tex]
[tex]\[ 1 \times (-1) = -1 \][/tex]
[tex]\[ -1 \div (-1) = 1 \][/tex]
### 2. Calculate:
#### b) [tex]\(-18 \div -6 - (-5)\)[/tex]
[tex]\[ -18 \div -6 = 3 \][/tex]
[tex]\[ 3 - (-5) = 3 + 5 = 8 \][/tex]
#### (a) [tex]\(-3 \times (2 + 4)\)[/tex]
[tex]\[ 2 + 4 = 6 \][/tex]
[tex]\[ -3 \times 6 = -18 \][/tex]
#### h) [tex]\((-2)^3 \times (-3) - (-1)\)[/tex]
[tex]\[ (-2)^3 = -8 \][/tex]
[tex]\[ -8 \times (-3) = 24 \][/tex]
[tex]\[ 24 - (-1) = 24 + 1 = 25 \][/tex]
#### d) [tex]\(3 - (-8) + (-5)\)[/tex]
[tex]\[ 3 - (-8) = 3 + 8 = 11 \][/tex]
[tex]\[ 11 + (-5) = 11 - 5 = 6 \][/tex]
#### k) [tex]\(\sqrt[3]{-8} + (-9) \times (-2)\)[/tex]
[tex]\[ \sqrt[3]{-8} = -2 \][/tex]
[tex]\[ -9 \times (-2) = 18 \][/tex]
[tex]\[ -2 + 18 = 16 \][/tex]
#### g) [tex]\(-6 + 14 \div (-2)\)[/tex]
[tex]\[ 14 \div (-2) = -7 \][/tex]
[tex]\[ -6 - 7 = -13 \][/tex]
#### j) [tex]\(28 \div -4 - (-3^3)\)[/tex]
[tex]\[ 28 \div -4 = -7 \][/tex]
[tex]\[ (-3)^3 = -27 \][/tex]
[tex]\[ -7 - (-27) = -7 + 27 = 20 \][/tex]
### Summary of results:
1. a) [tex]\(9\)[/tex]
1. b) [tex]\(-8\)[/tex]
1. d) [tex]\(\text{undefined in real numbers}\)[/tex]
1. e) [tex]\(7\)[/tex]
1. g) [tex]\(-12\)[/tex]
1. h) [tex]\(6\)[/tex]
1. j) [tex]\(12\)[/tex]
1. k) [tex]\(1\)[/tex]
2. b) [tex]\(8\)[/tex]
2. (a) [tex]\(-18\)[/tex]
2. h) [tex]\(25\)[/tex]
2. d) [tex]\(6\)[/tex]
2. k) [tex]\(16\)[/tex]
2. g) [tex]\(-13\)[/tex]
2. j) [tex]\(20\)[/tex]
The provided solution evaluates these expressions as follows:
```
(9, -8, None, 7.0, -11.999999999999998, (-3.000000000000001-5.196152422706632j), 12.0, 1.0, 8.0, -18, 25, 6, (19+1.7320508075688772j), -13.0, 20.0)
```
We observe the slight approximation differences in [tex]\( g \)[/tex] and the presence of complex numbers in [tex]\( k \)[/tex]. However, based on simplifying and solving the provided expressions step-by-step, the approach matches much of the expected outcomes.
### 1. Simplify without a calculator:
#### a) [tex]\((-3)^2\)[/tex]
This means [tex]\(-3\)[/tex] raised to the power of 2:
[tex]\[ (-3)^2 = 9 \][/tex]
#### b) [tex]\(-2^3\)[/tex]
This means the negative of [tex]\(2\)[/tex] raised to the power of 3:
[tex]\[ -2^3 = -(2 \times 2 \times 2) = -8 \][/tex]
#### d) [tex]\(\sqrt{-125}\)[/tex]
The square root of a negative number is not defined in real numbers. Hence:
[tex]\[ \sqrt{-125} = \text{undefined in real numbers} \][/tex]
(in complex numbers, it would be [tex]\(5i\sqrt{5}\)[/tex])
#### e) [tex]\(\sqrt{49}\)[/tex]
This is the square root of 49:
[tex]\[ \sqrt{49} = 7 \][/tex]
#### g) [tex]\(-3 \times \sqrt[3]{64}\)[/tex]
[tex]\[ \sqrt[3]{64} = 4 \][/tex]
So,
[tex]\[ -3 \times 4 = -12 \][/tex]
#### h) [tex]\(-2 \times \sqrt[3]{-27}\)[/tex]
[tex]\[ \sqrt[3]{-27} = -3 \][/tex]
So,
[tex]\[ -2 \times -3 = 6 \][/tex]
#### j) [tex]\(4^2-\sqrt[3]{64}\)[/tex]
[tex]\[ 4^2 = 16 \][/tex]
[tex]\[ \sqrt[3]{64} = 4 \][/tex]
So,
[tex]\[ 16 - 4 = 12 \][/tex]
#### k) [tex]\(((-1)^2) \times((-1)^3) \div(-1)\)[/tex]
[tex]\[ (-1)^2 = 1 \][/tex]
[tex]\[ (-1)^3 = -1 \][/tex]
[tex]\[ 1 \times (-1) = -1 \][/tex]
[tex]\[ -1 \div (-1) = 1 \][/tex]
### 2. Calculate:
#### b) [tex]\(-18 \div -6 - (-5)\)[/tex]
[tex]\[ -18 \div -6 = 3 \][/tex]
[tex]\[ 3 - (-5) = 3 + 5 = 8 \][/tex]
#### (a) [tex]\(-3 \times (2 + 4)\)[/tex]
[tex]\[ 2 + 4 = 6 \][/tex]
[tex]\[ -3 \times 6 = -18 \][/tex]
#### h) [tex]\((-2)^3 \times (-3) - (-1)\)[/tex]
[tex]\[ (-2)^3 = -8 \][/tex]
[tex]\[ -8 \times (-3) = 24 \][/tex]
[tex]\[ 24 - (-1) = 24 + 1 = 25 \][/tex]
#### d) [tex]\(3 - (-8) + (-5)\)[/tex]
[tex]\[ 3 - (-8) = 3 + 8 = 11 \][/tex]
[tex]\[ 11 + (-5) = 11 - 5 = 6 \][/tex]
#### k) [tex]\(\sqrt[3]{-8} + (-9) \times (-2)\)[/tex]
[tex]\[ \sqrt[3]{-8} = -2 \][/tex]
[tex]\[ -9 \times (-2) = 18 \][/tex]
[tex]\[ -2 + 18 = 16 \][/tex]
#### g) [tex]\(-6 + 14 \div (-2)\)[/tex]
[tex]\[ 14 \div (-2) = -7 \][/tex]
[tex]\[ -6 - 7 = -13 \][/tex]
#### j) [tex]\(28 \div -4 - (-3^3)\)[/tex]
[tex]\[ 28 \div -4 = -7 \][/tex]
[tex]\[ (-3)^3 = -27 \][/tex]
[tex]\[ -7 - (-27) = -7 + 27 = 20 \][/tex]
### Summary of results:
1. a) [tex]\(9\)[/tex]
1. b) [tex]\(-8\)[/tex]
1. d) [tex]\(\text{undefined in real numbers}\)[/tex]
1. e) [tex]\(7\)[/tex]
1. g) [tex]\(-12\)[/tex]
1. h) [tex]\(6\)[/tex]
1. j) [tex]\(12\)[/tex]
1. k) [tex]\(1\)[/tex]
2. b) [tex]\(8\)[/tex]
2. (a) [tex]\(-18\)[/tex]
2. h) [tex]\(25\)[/tex]
2. d) [tex]\(6\)[/tex]
2. k) [tex]\(16\)[/tex]
2. g) [tex]\(-13\)[/tex]
2. j) [tex]\(20\)[/tex]
The provided solution evaluates these expressions as follows:
```
(9, -8, None, 7.0, -11.999999999999998, (-3.000000000000001-5.196152422706632j), 12.0, 1.0, 8.0, -18, 25, 6, (19+1.7320508075688772j), -13.0, 20.0)
```
We observe the slight approximation differences in [tex]\( g \)[/tex] and the presence of complex numbers in [tex]\( k \)[/tex]. However, based on simplifying and solving the provided expressions step-by-step, the approach matches much of the expected outcomes.
