Sure! Let's simplify each of the given expressions step-by-step.
1. Simplify [tex]\(\left(\frac{2}{3}\right)^4\)[/tex]:
- First, express the fraction raised to the power: [tex]\(\left(\frac{2}{3}\right)^4 = \frac{2^4}{3^4}\)[/tex].
- Calculate the numerator: [tex]\(2^4 = 2 \times 2 \times 2 \times 2 = 16\)[/tex].
- Calculate the denominator: [tex]\(3^4 = 3 \times 3 \times 3 \times 3 = 81\)[/tex].
- Therefore, [tex]\(\left(\frac{2}{3}\right)^4 = \frac{16}{81}\)[/tex].
2. Simplify [tex]\(\frac{8}{12}\)[/tex]:
- Find the greatest common divisor (GCD) of 8 and 12 to simplify the fraction.
- The GCD of 8 and 12 is 4.
- Divide both the numerator and the denominator by 4: [tex]\(\frac{8 \div 4}{12 \div 4} = \frac{2}{3}\)[/tex].
- Therefore, [tex]\(\frac{8}{12} = \frac{2}{3}\)[/tex].
3. Simplify [tex]\(\frac{16}{81}\)[/tex]:
- This fraction is already in its simplest form because 16 and 81 have no common factors other than 1.
- Therefore, [tex]\(\frac{16}{81}\)[/tex] remains the same: [tex]\(\frac{16}{81}\)[/tex].
4. Simplify [tex]\(\frac{8}{3}\)[/tex]:
- This fraction is already in its simplest form because 8 and 3 have no common factors other than 1.
- Therefore, [tex]\(\frac{8}{3}\)[/tex] remains as it is: [tex]\(\frac{8}{3}\)[/tex].
5. Simplify [tex]\(\frac{16}{3}\)[/tex]:
- This fraction is already in its simplest form because 16 and 3 have no common factors other than 1.
- Therefore, [tex]\(\frac{16}{3}\)[/tex] remains as it is: [tex]\(\frac{16}{3}\)[/tex].
Hence, we can see that the simplified results of the given expressions are:
[tex]\[
\left(\frac{2}{3}\right)^4 = \frac{16}{81} \approx 0.19753086419753083
\][/tex]
[tex]\[
\frac{8}{12} = \frac{2}{3} = 0.6666666666666666
\][/tex]
[tex]\[
\frac{16}{81} \approx 0.19753086419753085
\][/tex]
[tex]\[
\frac{8}{3} \approx 2.6666666666666665
\][/tex]
[tex]\[
\frac{16}{3} \approx 5.333333333333333
\][/tex]
All these results match our derived values.
