The choices that are equivalent to the given expression are
A. [tex]\sqrt[4]{x^{9} }[/tex]
B. [tex](x^{9} )^{\frac{1}{4} }[/tex]
E. [tex](\sqrt[4]{x })^{9}[/tex]
From the question, we are to determine which of the given choices are equivalent to the given expression
The given expression is
[tex]x^{\frac{9}{4} }[/tex]
We will simplify each of the given choices to determine which are equivalent to the given expression
By law of indices,
[tex]\sqrt[n]{x^{m} } = x^{\frac{m}{n}}[/tex]
∴ [tex]\sqrt[4]{x^{9} }=x^{\frac{9}{4} }[/tex]
Hence, [tex]\sqrt[4]{x^{9} }[/tex] is equivalent to the given expression
If we clear the bracket and multiply the powers, we get
[tex]x^{9\times \frac{1}{4} }[/tex]
= [tex]x^{\frac{9}{4} }[/tex]
Hence, [tex](x^{9} )^{\frac{1}{4} }[/tex] is equivalent to the given expression
If we clear the bracket and multiply the powers, we get
[tex]x^{4\times \frac{1}{9} }[/tex]
= [tex]x^{\frac{4}{9} }[/tex]
Hence, [tex](x^{4} )^{\frac{1}{9} }[/tex] is not equivalent to the given expression
By law of indices,
[tex]\sqrt[n]{x^{m} } = x^{\frac{m}{n}}[/tex]
∴ [tex]\sqrt[9]{x^{4} }=x^{\frac{4}{9} }[/tex]
Hence, [tex]\sqrt[9]{x^{4} }[/tex] is not equivalent to the given expression
By law of indices,
[tex](\sqrt[n]{x })^{m}= (x^{\frac{1}{n}})^{m}[/tex]
∴ [tex](\sqrt[4]{x })^{9}= (x^{\frac{1}{4}})^{9}[/tex]
If we clear the bracket and multiply the powers, we get
[tex]x^{\frac{1}{4}\times 9 }[/tex]
= [tex]x^{\frac{9}{4} }[/tex]
Hence, [tex](\sqrt[4]{x })^{9}[/tex] is equivalent to the given expression
By law of indices,
[tex](\sqrt[n]{x })^{m}= (x^{\frac{1}{n}})^{m}[/tex]
∴ [tex](\sqrt[9]{x })^{4}= (x^{\frac{1}{9}})^{4}[/tex]
If we clear the bracket and multiply the powers, we get
[tex]x^{\frac{1}{9}\times 4 }[/tex]
= [tex]x^{\frac{4}{9} }[/tex]
Hence, [tex](\sqrt[9]{x })^{4}[/tex] is not equivalent to the given expression
Thus, the choices that are equivalent to the given expression are
A. [tex]\sqrt[4]{x^{9} }[/tex]
B. [tex](x^{9} )^{\frac{1}{4} }[/tex]
E. [tex](\sqrt[4]{x })^{9}[/tex]
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