Question

which expressions are equivalent to ( x^{\frac{4}{5}} )? choose all answers that apply: a ( (x^4)^5 ) b ( left( sqrt4{x}
ight)^5 ) c ( sqrt5{\frac{1}{x^4}} ) d none of the above

Explanation:

Step1: Recall Exponent Rules

Recall the rule for fractional exponents: \(a^{\frac{m}{n}}=\sqrt[n]{a^m}=(\sqrt[n]{a})^m\), and the power - of - a - power rule: \((a^m)^n = a^{m\times n}\). Also, recall the negative exponent rule: \(a^{-n}=\frac{1}{a^n}\).

Step2: Analyze Option A

For the expression \((x^{4})^{5}\), using the power - of - a - power rule \((a^m)^n=a^{m\times n}\), we have \((x^{4})^{5}=x^{4\times5}=x^{20}\), which is not equal to \(x^{\frac{4}{5}}\).

Step3: Analyze Option B

For the expression \((\sqrt[4]{x})^{5}\), since \(\sqrt[4]{x}=x^{\frac{1}{4}}\) (by the definition of roots and exponents, \(\sqrt[n]{a}=a^{\frac{1}{n}}\)), then \((\sqrt[4]{x})^{5}=(x^{\frac{1}{4}})^{5}\). Using the power - of - a - power rule \((a^m)^n = a^{m\times n}\), we get \((x^{\frac{1}{4}})^{5}=x^{\frac{1}{4}\times5}=x^{\frac{5}{4}}\), which is not equal to \(x^{\frac{4}{5}}\).

Step4: Analyze Option C

For the expression \(\sqrt[5]{\frac{1}{x^{4}}}\), we can rewrite \(\frac{1}{x^{4}}\) as \(x^{-4}\) (by the negative exponent rule \(a^{-n}=\frac{1}{a^n}\)). Then \(\sqrt[5]{\frac{1}{x^{4}}}=\sqrt[5]{x^{-4}}\). Since \(\sqrt[n]{a^m}=a^{\frac{m}{n}}\), we have \(\sqrt[5]{x^{-4}}=(x^{-4})^{\frac{1}{5}}\). Using the power - of - a - power rule \((a^m)^n=a^{m\times n}\), we get \((x^{-4})^{\frac{1}{5}}=x^{-4\times\frac{1}{5}}=x^{-\frac{4}{5}}=\frac{1}{x^{\frac{4}{5}}}\), which is not equal to \(x^{\frac{4}{5}}\).

Answer:

D. None of the above