To solve the expression [tex]\(\sqrt[3]{\frac{27}{8}} \div \sqrt[3]{1000}\)[/tex], we need to break it down into manageable steps. Here is the detailed, step-by-step solution:
1. Calculate the cube root of [tex]\(\frac{27}{8}\)[/tex]:
[tex]\[
\left(\frac{27}{8}\right)^{\frac{1}{3}}
\][/tex]
2. Find the cube root of 27 and 8 individually:
[tex]\[
\text{Cube root of } 27 = 3 \quad \text{(since } 3^3 = 27\text{)}
\][/tex]
[tex]\[
\text{Cube root of } 8 = 2 \quad \text{(since } 2^3 = 8\text{)}
\][/tex]
3. Combine the cube roots:
[tex]\[
\left(\frac{27}{8}\right)^{\frac{1}{3}} = \frac{3}{2} = 1.5
\][/tex]
4. Calculate the cube root of 1000:
[tex]\[
1000^{\frac{1}{3}}
\][/tex]
[tex]\[
\text{Cube root of } 1000 = 10 \quad \text{(since } 10^3 = 1000\text{)}
\][/tex]
5. Perform the division:
[tex]\[
1.5 \div 10 = 0.15
\][/tex]
Collectively, the values are:
- Cube root of [tex]\(\frac{27}{8}\)[/tex] is 1.5.
- Cube root of 1000 is approximated as 9.999999999999998 (which is essentially 10).
Therefore, the result of the division [tex]\(\sqrt[3]{\frac{27}{8}} \div \sqrt[3]{1000}\)[/tex] is:
[tex]\[
\frac{1.5}{9.999999999999998} \approx 0.15000000000000002
\][/tex]
So, the step-by-step solution yields the final result as approximately [tex]\(0.15\)[/tex].
