Certainly! Let's solve each part of the question step-by-step:
### 1.1.1 [tex]\( 118.01 \times 1000 \)[/tex]
To solve this problem, we need to multiply 118.01 by 1000.
[tex]\[ 118.01 \times 1000 = 118010.0 \][/tex]
### 1.1.2 [tex]\( 23\left(\frac{1}{3}+\frac{1}{6}\right) \)[/tex]
First, let's find a common denominator for the fractions inside the parentheses:
[tex]\[ \frac{1}{3} = \frac{2}{6} \][/tex]
Now add the fractions:
[tex]\[ \frac{1}{3} + \frac{1}{6} = \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2} \][/tex]
Next, multiply by 23:
[tex]\[ 23 \times \frac{1}{2} = \frac{23}{2} = 11.5 \][/tex]
### 1.1.3 [tex]\( 3 \sqrt{\frac{1}{65}} - (\sqrt{9})^2 \)[/tex]
First, simplify the terms individually:
[tex]\[ \sqrt{\frac{1}{65}} = \frac{1}{\sqrt{65}} \][/tex]
[tex]\[ \sqrt{9} = 3 \][/tex]
Now, substitute and simplify:
[tex]\[ 3 \times \frac{1}{\sqrt{65}} = \frac{3}{\sqrt{65}} \][/tex]
To rationalize the denominator:
[tex]\[ \frac{3}{\sqrt{65}} \times \frac{\sqrt{65}}{\sqrt{65}} = \frac{3 \sqrt{65}}{65} \][/tex]
Now, solving for [tex]\( 3 \sqrt{\frac{1}{65}} \)[/tex]:
[tex]\[ \frac{3}{\sqrt{65}} = 0.3721042037676254 \][/tex]
We know [tex]\( (\sqrt{9})^2 = 3^2 = 9 \)[/tex]:
[tex]\[ 0.3721042037676254 - 9 = -8.627895796232375 \][/tex]
### 1.1.4 [tex]\( \frac{1^4}{-3} \)[/tex]
To solve this, we first evaluate [tex]\( 1^4 \)[/tex]:
[tex]\[ 1^4 = 1 \][/tex]
Then divide by -3:
[tex]\[ \frac{1}{-3} = -0.3333333333333333 \][/tex]
In conclusion, we have the following results:
1. [tex]\( 118.01 \times 1000 = 118010.0 \)[/tex]
2. [tex]\( 23 \left( \frac{1}{3} + \frac{1}{6} \right) = 11.5 \)[/tex]
3. [tex]\( 3 \sqrt{\frac{1}{65}} - (\sqrt{9})^2 = -8.627895796232375 \)[/tex]
4. [tex]\( \frac{1^4}{-3} = -0.3333333333333333 \)[/tex]
These steps are how you arrive at the final answers for each part of the question.
