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Sagot :
To determine which of the given expressions are equivalent to \( x^{8/5} \), we need to express each of the choices in a similar form and compare them to \( x^{8/5} \).
Let's analyze each option step by step:
#### A. \( \left(x^3\right)^{2 / 5} \)
Applying the rule of exponents \( (a^m)^n = a^{m \cdot n} \):
[tex]\[ \left(x^3\right)^{2 / 5} = x^{3 \cdot \frac{2}{5}} = x^{\frac{6}{5}} \][/tex]
Since \( x^{\frac{6}{5}} \neq x^{\frac{8}{5}} \), this expression is not equivalent to \( x^{8 / 5} \).
#### B. \( \sqrt[8]{x^5} \)
Rewriting the radical expression as an exponent, \( \sqrt[n]{a} = a^{1/n} \):
[tex]\[ \sqrt[8]{x^5} = x^{\frac{5}{8}} \][/tex]
Since \( x^{\frac{5}{8}} \neq x^{\frac{8}{5}} \), this expression is not equivalent to \( x^{8 / 5} \).
#### C. \( (\sqrt[5]{x})^8 \)
Rewriting the radical expression as an exponent, \( \sqrt[n]{a} = a^{1/n} \):
[tex]\[ \sqrt[5]{x} = x^{\frac{1}{5}} \][/tex]
Thus,
[tex]\[ (\sqrt[5]{x})^8 = \left(x^{\frac{1}{5}}\right)^8 = x^{\frac{1}{5} \cdot 8} = x^{\frac{8}{5}} \][/tex]
Since \( x^{\frac{8}{5}} = x^{\frac{8}{5}} \), this expression is equivalent to \( x^{8 / 5} \).
#### D. \( \left(x^5\right)^{1 / 8} \)
Applying the rule of exponents \( (a^m)^n = a^{m \cdot n} \):
[tex]\[ \left(x^5\right)^{1 / 8} = x^{5 \cdot \frac{1}{8}} = x^{\frac{5}{8}} \][/tex]
Since \( x^{\frac{5}{8}} \neq x^{\frac{8}{5}} \), this expression is not equivalent to \( x^{8 / 5} \).
#### E. \( (\sqrt[8]{x})^5 \)
Rewriting the radical expression as an exponent, \( \sqrt[n]{a} = a^{1/n} \):
[tex]\[ \sqrt[8]{x} = x^{\frac{1}{8}} \][/tex]
Thus,
[tex]\[ (\sqrt[8]{x})^5 = \left(x^{\frac{1}{8}}\right)^5 = x^{\frac{1}{8} \cdot 5} = x^{\frac{5}{8}} \][/tex]
Since \( x^{\frac{5}{8}} \neq x^{\frac{8}{5}} \), this expression is not equivalent to \( x^{8 / 5} \).
#### F. \( \sqrt[5]{x^8} \)
Rewriting the radical expression as an exponent, \( \sqrt[n]{a^m} = (a^m)^{1/n} = a^{m/n} \):
[tex]\[ \sqrt[5]{x^8} = x^{\frac{8}{5}} \][/tex]
Since \( x^{\frac{8}{5}} = x^{\frac{8}{5}} \), this expression is equivalent to \( x^{8 / 5} \).
### Conclusion:
The expressions that are equivalent to \( x^{8 / 5} \) are:
- C. \( (\sqrt[5]{x})^8 \)
- F. \( \sqrt[5]{x^8} \)
Thus, the correct choices are:
[tex]\[ \text{C and F} \][/tex]
Let's analyze each option step by step:
#### A. \( \left(x^3\right)^{2 / 5} \)
Applying the rule of exponents \( (a^m)^n = a^{m \cdot n} \):
[tex]\[ \left(x^3\right)^{2 / 5} = x^{3 \cdot \frac{2}{5}} = x^{\frac{6}{5}} \][/tex]
Since \( x^{\frac{6}{5}} \neq x^{\frac{8}{5}} \), this expression is not equivalent to \( x^{8 / 5} \).
#### B. \( \sqrt[8]{x^5} \)
Rewriting the radical expression as an exponent, \( \sqrt[n]{a} = a^{1/n} \):
[tex]\[ \sqrt[8]{x^5} = x^{\frac{5}{8}} \][/tex]
Since \( x^{\frac{5}{8}} \neq x^{\frac{8}{5}} \), this expression is not equivalent to \( x^{8 / 5} \).
#### C. \( (\sqrt[5]{x})^8 \)
Rewriting the radical expression as an exponent, \( \sqrt[n]{a} = a^{1/n} \):
[tex]\[ \sqrt[5]{x} = x^{\frac{1}{5}} \][/tex]
Thus,
[tex]\[ (\sqrt[5]{x})^8 = \left(x^{\frac{1}{5}}\right)^8 = x^{\frac{1}{5} \cdot 8} = x^{\frac{8}{5}} \][/tex]
Since \( x^{\frac{8}{5}} = x^{\frac{8}{5}} \), this expression is equivalent to \( x^{8 / 5} \).
#### D. \( \left(x^5\right)^{1 / 8} \)
Applying the rule of exponents \( (a^m)^n = a^{m \cdot n} \):
[tex]\[ \left(x^5\right)^{1 / 8} = x^{5 \cdot \frac{1}{8}} = x^{\frac{5}{8}} \][/tex]
Since \( x^{\frac{5}{8}} \neq x^{\frac{8}{5}} \), this expression is not equivalent to \( x^{8 / 5} \).
#### E. \( (\sqrt[8]{x})^5 \)
Rewriting the radical expression as an exponent, \( \sqrt[n]{a} = a^{1/n} \):
[tex]\[ \sqrt[8]{x} = x^{\frac{1}{8}} \][/tex]
Thus,
[tex]\[ (\sqrt[8]{x})^5 = \left(x^{\frac{1}{8}}\right)^5 = x^{\frac{1}{8} \cdot 5} = x^{\frac{5}{8}} \][/tex]
Since \( x^{\frac{5}{8}} \neq x^{\frac{8}{5}} \), this expression is not equivalent to \( x^{8 / 5} \).
#### F. \( \sqrt[5]{x^8} \)
Rewriting the radical expression as an exponent, \( \sqrt[n]{a^m} = (a^m)^{1/n} = a^{m/n} \):
[tex]\[ \sqrt[5]{x^8} = x^{\frac{8}{5}} \][/tex]
Since \( x^{\frac{8}{5}} = x^{\frac{8}{5}} \), this expression is equivalent to \( x^{8 / 5} \).
### Conclusion:
The expressions that are equivalent to \( x^{8 / 5} \) are:
- C. \( (\sqrt[5]{x})^8 \)
- F. \( \sqrt[5]{x^8} \)
Thus, the correct choices are:
[tex]\[ \text{C and F} \][/tex]
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