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Which statement is true?​

Which statement is true?​-example-1
User Khio
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6 votes

Hello!

The answer is:

The second option,


(\sqrt[m]{x^(a) } )^(b)=\sqrt[m]{x^(ab) }

Why?

Discarding each given option in order to find the correct one, we have:

First option,


\sqrt[m]{x}\sqrt[m]{y}=\sqrt[2m]{xy}

The statement is false, the correct form of the statement (according to the property of roots) is:


\sqrt[m]{x}\sqrt[m]{y}=\sqrt[m]{xy}

Second option,


(\sqrt[m]{x^(a) } )^(b)=\sqrt[m]{x^(ab) }

The statement is true, we can prove it by using the following properties of exponents:


(a^(b))^(c)=a^(bc)


\sqrt[n]{x^(m) }=x^{(m)/(n) }

We are given the expression:


(\sqrt[m]{x^(a) } )^(b)

So, applying the properties, we have:


(\sqrt[m]{x^(a) } )^(b)=(x^{(a)/(m)})^(b)=x^{(ab)/(m)}\\\\x^{(ab)/(m)}=\sqrt[m]{x^(ab) }

Hence,


(\sqrt[m]{x^(a) } )^(b)=\sqrt[m]{x^(ab) }

Third option,


a\sqrt[n]{x}+b\sqrt[n]{x}=ab\sqrt[n]{x}

The statement is false, the correct form of the statement (according to the property of roots) is:


a\sqrt[n]{x}+b\sqrt[n]{x}=(a+b)\sqrt[n]{x}

Fourth option,


\frac{\sqrt[m]{x} }{\sqrt[m]{y}}=m√(xy)

The statement is false, the correct form of the statement (according to the property of roots) is:


\frac{\sqrt[m]{x} }{\sqrt[m]{y}}=\sqrt[m]{(x)/(y) }

Hence, the answer is, the statement that is true is the second statement:


(\sqrt[m]{x^(a) } )^(b)=\sqrt[m]{x^(ab) }

Have a nice day!

User DDovzhenko
by
8.5k points

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