Answer :
Let's find the cube roots of the given fractions step by step.
### a) [tex]\( \frac{8}{27} \)[/tex]
To find the cube root of [tex]\( \frac{8}{27} \)[/tex], we'll take the cube root of the numerator and the cube root of the denominator separately.
- The cube root of 8:
[tex]\[ \sqrt[3]{8} = 2 \][/tex]
- The cube root of 27:
[tex]\[ \sqrt[3]{27} = 3 \][/tex]
Thus, the cube root of [tex]\( \frac{8}{27} \)[/tex] is:
[tex]\[ \frac{\sqrt[3]{8}}{\sqrt[3]{27}} = \frac{2}{3} \][/tex]
### b) [tex]\( \frac{64}{125} \)[/tex]
Next, for [tex]\( \frac{64}{125} \)[/tex]:
- The cube root of 64:
[tex]\[ \sqrt[3]{64} = 4 \][/tex]
- The cube root of 125:
[tex]\[ \sqrt[3]{125} = 5 \][/tex]
Thus, the cube root of [tex]\( \frac{64}{125} \)[/tex] is:
[tex]\[ \frac{\sqrt[3]{64}}{\sqrt[3]{125}} = \frac{4}{5} \][/tex]
### c) [tex]\( \frac{125}{216} \)[/tex]
Now, for [tex]\( \frac{125}{216} \)[/tex]:
- The cube root of 125:
[tex]\[ \sqrt[3]{125} = 5 \][/tex]
- The cube root of 216:
[tex]\[ \sqrt[3]{216} = 6 \][/tex]
Thus, the cube root of [tex]\( \frac{125}{216} \)[/tex] is:
[tex]\[ \frac{\sqrt[3]{125}}{\sqrt[3]{216}} = \frac{5}{6} \][/tex]
### d) [tex]\( \frac{343}{1728} \)[/tex]
Finally, for [tex]\( \frac{343}{1728} \)[/tex]:
- The cube root of 343:
[tex]\[ \sqrt[3]{343} = 7 \][/tex]
- The cube root of 1728:
[tex]\[ \sqrt[3]{1728} = 12 \][/tex]
Thus, the cube root of [tex]\( \frac{343}{1728} \)[/tex] is:
[tex]\[ \frac{\sqrt[3]{343}}{\sqrt[3]{1728}} = \frac{7}{12} \][/tex]
In conclusion, the cube roots of the given fractions are:
a) [tex]\( \frac{2}{3} \)[/tex]
b) [tex]\( \frac{4}{5} \)[/tex]
c) [tex]\( \frac{5}{6} \)[/tex]
d) [tex]\( \frac{7}{12} \)[/tex]
### a) [tex]\( \frac{8}{27} \)[/tex]
To find the cube root of [tex]\( \frac{8}{27} \)[/tex], we'll take the cube root of the numerator and the cube root of the denominator separately.
- The cube root of 8:
[tex]\[ \sqrt[3]{8} = 2 \][/tex]
- The cube root of 27:
[tex]\[ \sqrt[3]{27} = 3 \][/tex]
Thus, the cube root of [tex]\( \frac{8}{27} \)[/tex] is:
[tex]\[ \frac{\sqrt[3]{8}}{\sqrt[3]{27}} = \frac{2}{3} \][/tex]
### b) [tex]\( \frac{64}{125} \)[/tex]
Next, for [tex]\( \frac{64}{125} \)[/tex]:
- The cube root of 64:
[tex]\[ \sqrt[3]{64} = 4 \][/tex]
- The cube root of 125:
[tex]\[ \sqrt[3]{125} = 5 \][/tex]
Thus, the cube root of [tex]\( \frac{64}{125} \)[/tex] is:
[tex]\[ \frac{\sqrt[3]{64}}{\sqrt[3]{125}} = \frac{4}{5} \][/tex]
### c) [tex]\( \frac{125}{216} \)[/tex]
Now, for [tex]\( \frac{125}{216} \)[/tex]:
- The cube root of 125:
[tex]\[ \sqrt[3]{125} = 5 \][/tex]
- The cube root of 216:
[tex]\[ \sqrt[3]{216} = 6 \][/tex]
Thus, the cube root of [tex]\( \frac{125}{216} \)[/tex] is:
[tex]\[ \frac{\sqrt[3]{125}}{\sqrt[3]{216}} = \frac{5}{6} \][/tex]
### d) [tex]\( \frac{343}{1728} \)[/tex]
Finally, for [tex]\( \frac{343}{1728} \)[/tex]:
- The cube root of 343:
[tex]\[ \sqrt[3]{343} = 7 \][/tex]
- The cube root of 1728:
[tex]\[ \sqrt[3]{1728} = 12 \][/tex]
Thus, the cube root of [tex]\( \frac{343}{1728} \)[/tex] is:
[tex]\[ \frac{\sqrt[3]{343}}{\sqrt[3]{1728}} = \frac{7}{12} \][/tex]
In conclusion, the cube roots of the given fractions are:
a) [tex]\( \frac{2}{3} \)[/tex]
b) [tex]\( \frac{4}{5} \)[/tex]
c) [tex]\( \frac{5}{6} \)[/tex]
d) [tex]\( \frac{7}{12} \)[/tex]
