If [tex]$a^{1 / n}=r$[/tex], then which of the following are true statements?
Check all that apply.
A. [tex]$r^n=a$[/tex]
B. [tex]$\sqrt[n]{a}=r$[/tex]
C. [tex]$n^{1 / r}=a$[/tex]
D. [tex]$a^r=n$[/tex]
Answer (1)
Given a 1/ n = r , raise both sides to the power of n to show that r n = a .
Recognize that a 1/ n is equivalent to n a , thus n a = r .
Provide counterexamples to demonstrate that n 1/ r = a and a r = n are not necessarily true.
Conclude that the true statements are A and B, so the answer is A , B .
Explanation
Understanding the Problem We are given the equation a 1/ n = r and asked to determine which of the following statements are true: A. r n = a B. n a = r C. n 1/ r = a D. a r = n
Analyzing Statement A Statement A: r n = a . To check if this is true, we start with the given equation a 1/ n = r . Raising both sides to the power of n , we get ( a 1/ n ) n = r n . Since ( a 1/ n ) n = a , we have a = r n . Thus, statement A is true.
Analyzing Statement B Statement B: n a = r . The expression a 1/ n is equivalent to n a . Since we are given a 1/ n = r , it follows that n a = r . Thus, statement B is true.
Analyzing Statement C Statement C: n 1/ r = a . We are given a 1/ n = r . There is no direct algebraic manipulation to transform a 1/ n = r into n 1/ r = a . To see that this is not necessarily true, we can find a counterexample. Let a = 4 and n = 2 . Then a 1/ n = 4 1/2 = 2 , so r = 2 . Statement C becomes 2 1/2 = 4 , which is false. Thus, statement C is not necessarily true.
Analyzing Statement D Statement D: a r = n . We are given a 1/ n = r . There is no direct algebraic manipulation to transform a 1/ n = r into a r = n . To see that this is not necessarily true, we can find a counterexample. Let a = 4 and n = 2 . Then a 1/ n = 4 1/2 = 2 , so r = 2 . Statement D becomes 4 2 = 2 , which is false. Thus, statement D is not necessarily true.
Conclusion Therefore, the true statements are A and B.
Examples
Understanding exponents and roots is crucial in various fields, such as finance when calculating compound interest or in physics when dealing with exponential decay. For instance, if an investment grows at a rate such that its value is multiplied by 1.1 every year, then after n years, the initial investment P will be P × ( 1.1 ) n . Conversely, if we want to find out how many years it takes for the investment to double, we would need to solve for n in the equation 2 = ( 1.1 ) n , which involves understanding the relationship between exponents and roots.
