Question

"What is a counterexample for the statement ‘If
`x^(4)` equals 81, then x must be 3’?

A) x must be -3
B) x must be 2
C) x must be -4
D) x must be 9"

Please select the option that serves as a counterexample to the statement.

Answer 1

A counterexample to 'If x^4 equals 81, then x must be 3' is A) x must be -3 because (-3)^4 also equals 81, showing that x can be negative and still satisfy the equation.

Step-by-step explanation:

The statement you're asking about is 'If x4 equals 81, then x must be 3'. The validity of this statement can be tested by finding a counterexample. A counterexample is a specific case which shows that the statement is not always true.

To find a counterexample, we need to think of a different number whose fourth power also equals 81. Since 81 is a positive number, we should remember that raising both positive and negative numbers to an even power will result in a positive number. The number -3 is a suitable choice because (-3)4 = 81 as well. Therefore, -3 serves as a valid counterexample because it shows that x can also be negative and still satisfy the condition x4 = 81.

In this case, the correct answer would be: A) x must be -3, which disproves the original statement by providing an example where x is not 3 but the equation x4 = 81 is still correct.