Hello!
The answer is:
The second option,
[tex](\sqrt[m]{x^{a} } )^{b}=\sqrt[m]{x^{ab} }[/tex]
Why?
Discarding each given option in order to find the correct one, we have:
First option,
[tex]\sqrt[m]{x}\sqrt[m]{y}=\sqrt[2m]{xy}[/tex]
The statement is false, the correct form of the statement (according to the property of roots) is:
[tex]\sqrt[m]{x}\sqrt[m]{y}=\sqrt[m]{xy}[/tex]
Second option,
[tex](\sqrt[m]{x^{a} } )^{b}=\sqrt[m]{x^{ab} }[/tex]
The statement is true, we can prove it by using the following properties of exponents:
[tex](a^{b})^{c}=a^{bc}[/tex]
[tex]\sqrt[n]{x^{m} }=x^{\frac{m}{n} }[/tex]
We are given the expression:
[tex](\sqrt[m]{x^{a} } )^{b}[/tex]
So, applying the properties, we have:
[tex](\sqrt[m]{x^{a} } )^{b}=(x^{\frac{a}{m}})^{b}=x^{\frac{ab}{m}}\\\\x^{\frac{ab}{m}}=\sqrt[m]{x^{ab} }[/tex]
Hence,
[tex](\sqrt[m]{x^{a} } )^{b}=\sqrt[m]{x^{ab} }[/tex]
Third option,
[tex]a\sqrt[n]{x}+b\sqrt[n]{x}=ab\sqrt[n]{x}[/tex]
The statement is false, the correct form of the statement (according to the property of roots) is:
[tex]a\sqrt[n]{x}+b\sqrt[n]{x}=(a+b)\sqrt[n]{x}[/tex]
Fourth option,
[tex]\frac{\sqrt[m]{x} }{\sqrt[m]{y}}=m\sqrt{xy}[/tex]
The statement is false, the correct form of the statement (according to the property of roots) is:
[tex]\frac{\sqrt[m]{x} }{\sqrt[m]{y}}=\sqrt[m]{\frac{x}{y} }[/tex]
Hence, the answer is, the statement that is true is the second statement:
[tex](\sqrt[m]{x^{a} } )^{b}=\sqrt[m]{x^{ab} }[/tex]
Have a nice day!
