Which of the following are true, given that x is positive?

Question

I.  \sqrt {x^n}=(\sqrt {x})^n

II.  x^{a^{b}}=x^{ab}

III.  x^{a+a+a+....a times}=x^{a^{2}}

A. I and II
B. II and III
C. I and III
D. I,II, and III
E. None the above

gmatophobia · Answer

Given  : x is positive

I.  \sqrt {x^n}=(\sqrt {x})^n

(x^n)^{\frac{1}{m}} = (x^{\frac{1}{m}})^{n}

x^{\frac{n}{m}} = x^{\frac{n}{m}}

This statement is true

II.  x^{a^{b}}=x^{ab}

Assume: x = 2, a = 3 & b = 2

2^{3^2} = 2^{9}

2^{3*6} = 2^{6}

2^{9} 
eq 2^{6}

This statement is not true.

III.  x^{a+a+a+....a times}=x^{a^{2}}

x^{a+a+a+....a 	ext{times}}

x^{a*a}=x^{a^{2}}

This statement is true

Option C

ahmedtnvr · Answer

Can someone please explain in detail why II is incorrect.

Bunuel · Answer

Remember the rule for working with stacked exponents, which is to begin with the highest exponent and work our way down. For example,  a^{m^n}  means we first compute  m^n  and then use that result as the exponent for  a . Therefore,  a^{m^n} = a^{(m^n)} . On the other hand, if we have  (a^m)^n , we compute  a^m  first and then raise the result to the power of  n , so  (a^m)^n=a^{mn} .

OutlawModric · Answer

For the third statement - what were to happend if a<0? Lets say if a is -2, Then this willl be invalid. Are we to assume that since it is said that a is "a" times, then a must be a positive number?

Which of the following are true, given that x is positive?

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Which of the following are true, given that x is positive?

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