Evaluate the expression. [tex]$\sqrt[18]{(-6)^{18}}$[/tex]
Answer (1)
Recognize that n x n = ∣ x ∣ when n is even.
Apply this property to the given expression with n = 18 and x = − 6 .
Therefore, 18 ( − 6 ) 18 = ∣ − 6∣ .
Evaluate the absolute value of -6, which is 6, so the final answer is 6 .
Explanation
Understanding the Problem We are asked to evaluate the expression 18 ( − 6 ) 18 . This involves finding the 18th root of ( − 6 ) raised to the 18th power. Since the index of the root is even (18), we need to consider the absolute value of the base.
Applying the Property of Even Roots Recall that n x n = ∣ x ∣ when n is an even number. In our case, n = 18 and x = − 6 . Therefore, we have 18 ( − 6 ) 18 = ∣ − 6∣ .
Evaluating the Absolute Value The absolute value of a number is its distance from zero. Thus, the absolute value of − 6 is 6. So, ∣ − 6∣ = 6 .
Final Answer Therefore, the expression 18 ( − 6 ) 18 evaluates to 6.
Examples
Consider a scenario where you are calculating the side length of a square given its area. If the area is represented by an expression involving an even power, taking the square root (or any even root) will give you the absolute value of the side length. For example, if the area is ( − x ) 2 , the side length would be ( − x ) 2 = ∣ x ∣ , ensuring the side length is positive regardless of the value of x . This concept is crucial in various fields like geometry, physics, and engineering where physical quantities must be positive.
