Answer :
We first analyze the equation
[tex]$$
f^2 = 49.
$$[/tex]
To solve this equation, we write it as
[tex]$$
f^2 - 49 = 0.
$$[/tex]
This can be factored as a difference of squares:
[tex]$$
(f - 7)(f + 7) = 0.
$$[/tex]
Setting each factor equal to zero gives:
[tex]$$
f - 7 = 0 \quad \text{or} \quad f + 7 = 0.
$$[/tex]
Thus, the solutions are
[tex]$$
f = 7 \quad \text{or} \quad f = -7.
$$[/tex]
So, both [tex]$7$[/tex] and [tex]$-7$[/tex] are valid values of [tex]$f$[/tex] for the equation [tex]$f^2 = 49$[/tex].
Next, consider the equation
[tex]$$
f^3 = 343.
$$[/tex]
Since [tex]$343 = 7^3$[/tex], we take the cube root of both sides:
[tex]$$
f = \sqrt[3]{343} = 7.
$$[/tex]
Notice that if we try [tex]$f = -7$[/tex], then
[tex]$$
(-7)^3 = -343 \neq 343,
$$[/tex]
which means [tex]$f = -7$[/tex] is not a solution for this equation.
Only the first equation has both [tex]$7$[/tex] and [tex]$-7$[/tex] as possible values for [tex]$f$[/tex]. Therefore, the correct answer is option A.
[tex]$$
f^2 = 49.
$$[/tex]
To solve this equation, we write it as
[tex]$$
f^2 - 49 = 0.
$$[/tex]
This can be factored as a difference of squares:
[tex]$$
(f - 7)(f + 7) = 0.
$$[/tex]
Setting each factor equal to zero gives:
[tex]$$
f - 7 = 0 \quad \text{or} \quad f + 7 = 0.
$$[/tex]
Thus, the solutions are
[tex]$$
f = 7 \quad \text{or} \quad f = -7.
$$[/tex]
So, both [tex]$7$[/tex] and [tex]$-7$[/tex] are valid values of [tex]$f$[/tex] for the equation [tex]$f^2 = 49$[/tex].
Next, consider the equation
[tex]$$
f^3 = 343.
$$[/tex]
Since [tex]$343 = 7^3$[/tex], we take the cube root of both sides:
[tex]$$
f = \sqrt[3]{343} = 7.
$$[/tex]
Notice that if we try [tex]$f = -7$[/tex], then
[tex]$$
(-7)^3 = -343 \neq 343,
$$[/tex]
which means [tex]$f = -7$[/tex] is not a solution for this equation.
Only the first equation has both [tex]$7$[/tex] and [tex]$-7$[/tex] as possible values for [tex]$f$[/tex]. Therefore, the correct answer is option A.
