To find the cube of a rational number [tex]\(\frac{a}{b}\)[/tex], follow these steps:
1. Understand the concept of cubing a fraction:
- The cube of [tex]\(\frac{a}{b}\)[/tex] is given by:
[tex]\[
\left( \frac{a}{b} \right)^3 = \frac{a^3}{b^3}
\][/tex]
2. Cube the numerator and the denominator separately:
- For the rational number [tex]\(\frac{a}{b}\)[/tex], raise both the numerator [tex]\(a\)[/tex] and the denominator [tex]\(b\)[/tex] to the power of 3.
Now, let's apply this method to the given rational numbers:
### For [tex]\(\left( \frac{2}{9} \right)^3\)[/tex]:
1. Identify the numerator and denominator:
- Numerator = 2
- Denominator = 9
2. Cube the numerator and the denominator:
- [tex]\(2^3 = 8\)[/tex]
- [tex]\(9^3 = 729\)[/tex]
3. Form the final expression:
- The cube of [tex]\(\frac{2}{9}\)[/tex] is:
[tex]\[
\left( \frac{2}{9} \right)^3 = \frac{8}{729} \approx 0.010973936899862825
\][/tex]
### For [tex]\(\left( \frac{-3}{8} \right)^3\)[/tex]:
1. Identify the numerator and denominator:
- Numerator = -3
- Denominator = 8
2. Cube the numerator and the denominator:
- [tex]\((-3)^3 = -27\)[/tex]
- [tex]\(8^3 = 512\)[/tex]
3. Form the final expression:
- The cube of [tex]\(\frac{-3}{8}\)[/tex] is:
[tex]\[
\left( \frac{-3}{8} \right)^3 = \frac{-27}{512} \approx -0.052734375
\][/tex]
### Conclusion:
- The value of [tex]\(\left( \frac{2}{9} \right)^3\)[/tex] is approximately 0.010973936899862825.
- The value of [tex]\(\left( \frac{-3}{8} \right)^3\)[/tex] is approximately -0.052734375.
