To simplify the expression \( \left( \frac{2b}{c} \right)^{3} \), we need to apply the power of a quotient rule. This rule states that when you raise a fraction to a power, you can apply the exponent to both the numerator and the denominator separately.
1. Start with the expression:
\( \left( \frac{2b}{c} \right)^{3} \)
2. Apply the exponent to both the numerator and the denominator:
\( = \frac{(2b)^{3}}{c^{3}} \)
3. Now, simplify the numerator:
\( (2b)^{3} = 2^{3} \cdot b^{3} = 8b^{3} \)
4. Therefore, we can rewrite the entire expression:
\( = \frac{8b^{3}}{c^{3}} \)
So, the correct simplification of the expression \( \left( \frac{2b}{c} \right)^{3} \) is:
\( \frac{8b^{3}}{c^{3}} \)