Sure, let's solve the problem step by step. We need to determine the value of [tex]\(\left(\frac{54}{250}\right)^{1/3}\)[/tex], and compare it with the given options.
1. Simplification of the Fraction:
First, we simplify the fraction [tex]\(\frac{54}{250}\)[/tex].
- Factorize both the numerator and the denominator:
[tex]\[
\frac{54}{250} = \frac{54 \div 2}{250 \div 2} = \frac{27}{125}
\][/tex]
2. Expression under Cube Root:
Now we need to find the cube root of the simplified fraction:
[tex]\[
\left(\frac{27}{125}\right)^{1/3}
\][/tex]
3. Cube Root of Numerator and Denominator:
We take the cube root separately for both the numerator and the denominator:
[tex]\[
\left(\frac{27}{125}\right)^{1/3} = \frac{27^{1/3}}{125^{1/3}}
\][/tex]
4. Evaluate Cube Roots:
- The cube root of 27:
[tex]\[
27^{1/3} = 3 \quad \text{(since } 3^3 = 27 \text{)}
\][/tex]
- The cube root of 125:
[tex]\[
125^{1/3} = 5 \quad \text{(since } 5^3 = 125 \text{)}
\][/tex]
5. Solution:
Combining these results:
[tex]\[
\frac{27^{1/3}}{125^{1/3}} = \frac{3}{5}
\][/tex]
Therefore, the value of [tex]\(\left(\frac{54}{250}\right)^{1/3}\)[/tex] is [tex]\(\frac{3}{5}\)[/tex].
So, the correct answer is:
[tex]\[
\boxed{\frac{3}{5}}
\][/tex]
