Sure, I will guide you step-by-step through each calculation.
### 2.2 Calculate without the use of a calculator:
#### 2.2.1 [tex]\( 7.5 + 0.75 + 0.075 + 75 \)[/tex]
First, let's align the decimal points and add these numbers carefully:
```
7.500
+ 0.750
+ 0.075
+75.000
--------
83.325
```
So, the sum of [tex]\( 7.5 + 0.75 + 0.075 + 75 \)[/tex] is [tex]\( 83.325 \)[/tex].
#### 2.2.2 [tex]\( 14.4 \div 0.12 \)[/tex]
To perform this division without a calculator, first convert the divisor ([tex]\( 0.12 \)[/tex]) to a whole number by multiplying both the numerator and the denominator by 100:
[tex]\[ \frac{14.4}{0.12} = \frac{14.4 \times 100}{0.12 \times 100} = \frac{1440}{12} \][/tex]
Now, perform the division:
[tex]\[ 1440 \div 12 = 120 \][/tex]
So, [tex]\( 14.4 \div 0.12 \)[/tex] equals [tex]\( 120 \)[/tex].
#### 2.2.3 [tex]\( \sqrt[3]{0.001} \)[/tex]
The cube root of [tex]\( 0.001 \)[/tex] is a number which when multiplied by itself three times gives [tex]\( 0.001 \)[/tex]. Recognizing that [tex]\( 0.001 \)[/tex] is [tex]\( 1 \times 10^{-3} \)[/tex], we find that:
[tex]\[ 0.001^{\frac{1}{3}} = (1 \times 10^{-3})^{\frac{1}{3}} = 1 \times 10^{-1} = 0.1 \][/tex]
So, the cube root of [tex]\( 0.001 \)[/tex] is [tex]\( 0.1 \)[/tex].
#### 2.2.4 [tex]\( (-0.4)^3 \)[/tex]
To find [tex]\( (-0.4)^3 \)[/tex], multiply [tex]\( -0.4 \)[/tex] by itself three times:
[tex]\[ (-0.4) \times (-0.4) = 0.16 \][/tex]
[tex]\[ 0.16 \times (-0.4) = -0.064 \][/tex]
So, [tex]\( (-0.4)^3 \)[/tex] is [tex]\( -0.064 \)[/tex].
### Summary of Results
- [tex]\( 7.5 + 0.75 + 0.075 + 75 = 83.325 \)[/tex]
- [tex]\( 14.4 \div 0.12 = 120 \)[/tex]
- [tex]\( \sqrt[3]{0.001} = 0.1 \)[/tex]
- [tex]\( (-0.4)^3 = -0.064 \)[/tex]
There you have the detailed step-by-step solutions to the given calculations!
