Question

Complete the following statement: |a| = |a2| if and only if |a| . . . .

Answer 1

The statement |a| = |a^2| is true if and only if a and a^2 have the same absolute value.

Step-by-step explanation:

The statement |a| = |a^2| is true if and only if |a| is equal to the absolute value of a^2. In other words, |a| = |a^2| if and only if a and a^2 have the same absolute value.

For example, if a = 2, then |a| = |2| = 2 and |a^2| = |4| = 4. Since 2 and 4 have the same absolute value, the statement is true.

However, if a = -3, then |a| = |-3| = 3 and |a^2| = |(-3)^2| = |9| = 9. Since 3 and 9 do not have the same absolute value, the statement is false.

Answer 2

The statement |a| = |a^2| is true if and only if the number a is a non-negative number that is either 0 or 1, as these are the only cases where a number equals its own square.

Step-by-step explanation:

Complete the following statement: |a| = |a2| if and only if |a| is a non-negative number. This is because the absolute value of any real number is always non-negative, and the square of any real number is also non-negative, making the absolute values of both equal. For example, if a = -3, then |a| = |-3| = 3 and a2 = (-3)2 = 9, so |a2| = |9| = 9. However, since |a| = 3 and not 9 in this case, they are not equal. Therefore, |a| = |a2| is only true when a >= 0. This can be further illustrated with a positive example: if a = 2, then |a| = |2| = 2 and |a2| = |22| = |4| = 4, which are not equal as well. Thus, the following condition must be met for the statement to hold: a must be either 0 or 1, the two non-negative numbers whose square equals the number itself.