Introduction:
In this problem, we are asked to find the number by which (-49) ^ -1 should be divided to obtain a quotient equal to (-7) ^ -1. To solve this problem, we will first calculate the values of (-49) ^ -1 and (-7) ^ -1, and then find the quotient.

Calculating (-49) ^ -1:
To calculate the value of (-49) ^ -1, we need to find the reciprocal of (-49). The reciprocal of a number is obtained by dividing 1 by that number. Therefore, the reciprocal of (-49) can be written as 1 / (-49).

Calculating (-7) ^ -1:
Similarly, we can calculate the value of (-7) ^ -1 by finding the reciprocal of (-7), which is 1 / (-7).

Finding the quotient:
Now, we can find the quotient by dividing the value of (-49) ^ -1 by the value of (-7) ^ -1. Mathematically, this can be expressed as (1 / (-49)) / (1 / (-7)).

Simplifying the quotient:
To simplify the quotient, we can multiply the numerator and denominator by the reciprocal of the denominator. In this case, the reciprocal of (1 / (-7)) is (-7 / 1). Therefore, the quotient becomes (1 / (-49)) * (-7 / 1).

Multiplying the fractions:
To multiply the fractions, we multiply the numerators and denominators separately. The numerator becomes 1 * (-7) = -7, and the denominator becomes (-49) * 1 = -49. Therefore, the quotient is -7 / -49.

Simplifying the final quotient:
To simplify the final quotient, we can divide both the numerator and denominator by their greatest common divisor (GCD), which is 7. Dividing -7 by 7 gives us -1, and dividing -49 by 7 gives us -7. Therefore, the simplified quotient is -1 / -7, which is equal to 1 / 7.

Conclusion:
By dividing (-49) ^ -1 by 1 / 7, we obtain a quotient equal to (-7) ^ -1. Therefore, the number by which (-49) ^ -1 should be divided is 1 / 7.

Let the number be x
(–49)–1 ÷ x = (–7)–1
⇒
1
49
  −
    ÷ x =
1
7
      −
⇒
–1 1
×
49 x =
1
−7
⇒ 49 x = 7 ⇒ x =
7
49
=
1
7
⋅