To evaluate the expression [tex]\(\sqrt[3]{\frac{1.9 \times 0.032 \times 0.08}{20 \times 0.0038}}\)[/tex] without using a table or calculator, we can break down the problem into smaller, straightforward steps. Here is a step-by-step solution:
### Step 1: Calculate the Numerator
First, we need to multiply the numbers in the numerator:
[tex]\[ 1.9 \times 0.032 \times 0.08 \][/tex]
Performing the multiplication sequentially:
[tex]\[ 1.9 \times 0.032 = 0.0608 \][/tex]
[tex]\[ 0.0608 \times 0.08 = 0.004864 \][/tex]
So, the numerator is:
[tex]\[ 0.004864 \][/tex]
### Step 2: Calculate the Denominator
Next, we need to multiply the numbers in the denominator:
[tex]\[ 20 \times 0.0038 \][/tex]
Performing the multiplication:
[tex]\[ 20 \times 0.0038 = 0.076 \][/tex]
So, the denominator is:
[tex]\[ 0.076 \][/tex]
### Step 3: Divide the Numerator by the Denominator
Now, we divide the result from the numerator by the result from the denominator:
[tex]\[ \frac{0.004864}{0.076} \][/tex]
Performing the division:
[tex]\[ \frac{0.004864}{0.076} = 0.064 \][/tex]
### Step 4: Calculate the Cube Root
Finally, we need to calculate the cube root of the result from the division:
[tex]\[ \sqrt[3]{0.064} \][/tex]
Recall that the cube root of 0.064 is:
[tex]\[ \sqrt[3]{0.064} = 0.4 \][/tex]
Thus, the evaluated expression is:
[tex]\[
\sqrt[3]{\frac{1.9 \times 0.032 \times 0.08}{20 \times 0.0038}} = 0.4
\][/tex]
So, the final answer is:
[tex]\[ 0.4 \][/tex]
