Which of the following is the radical expression of ${ }_{a^a}$ ?
$4 a^9$
$9 a^4$
$\sqrt[4]{a^9}$
$\sqrt[9]{a^4}$
Answer (1)
The given expression a a is likely a typo and should be interpreted as a 9 4 or a 4 9 .
If the expression is a 9 4 , the equivalent radical expression is 9 a 4 .
If the expression is a 4 9 , the equivalent radical expression is 4 a 9 .
Comparing with the given options, the most likely answer is 9 a 4 .
Explanation
Understanding the Problem We are given the expression a a and asked to find its equivalent radical expression from the options: 4 a 9 , 9 a 4 , 4 a 9 , 9 a 4 . It seems there is a typo in the original expression. It is likely that the intended expression was a 9 4 or a 4 9 . We will analyze both possibilities. Recall that a n m = n a m .
Case 1: a 9 4 If the expression is a 9 4 , then the equivalent radical expression is 9 a 4 . This is because the denominator of the fractional exponent becomes the index of the radical, and the numerator becomes the exponent of the radicand.
Case 2: a 4 9 If the expression is a 4 9 , then the equivalent radical expression is 4 a 9 . This is because the denominator of the fractional exponent becomes the index of the radical, and the numerator becomes the exponent of the radicand.
Finding the Correct Option Comparing the derived radical expressions with the given options, we see that 4 a 9 and 9 a 4 are among the options. Since the expression a a is most likely a typo and should be interpreted as either a 9 4 or a 4 9 , we choose the option that matches one of these interpretations.
Selecting the Most Likely Answer The options are: 4 a 9 9 a 4 4 a 9 9 a 4
We see that 4 a 9 and 9 a 4 are possible radical expressions. Without further information, we cannot definitively determine which one is correct. However, since the expression is written as a a , it is more likely that the intended expression is a 9 4 , which corresponds to 9 a 4 .
Final Answer Therefore, the most likely radical expression for a a is 9 a 4 .
Examples
Radical expressions are used in various fields, such as physics and engineering, to simplify complex equations and represent physical quantities. For example, the period of a simple pendulum can be expressed using a radical expression involving the length of the pendulum and the acceleration due to gravity. Similarly, in electrical engineering, the impedance of a circuit can be represented using radical expressions involving resistance and reactance. Understanding radical expressions allows us to manipulate and solve these equations more easily, leading to a better understanding of the underlying physical phenomena. Moreover, in computer graphics, radical expressions are used to calculate distances and perform transformations, enabling the creation of realistic images and animations.
