Alright, let's solve the problem step-by-step.
Given the equation:
[tex]\[ m^2 + \frac{1}{m^2} = 9 \][/tex]
We want to find the value of [tex]\( m - \frac{1}{m} \)[/tex].
We begin by introducing a new variable:
[tex]\[ x = m - \frac{1}{m} \][/tex]
Next, we need to relate [tex]\( x^2 \)[/tex] to the given equation. By squaring both sides of [tex]\( x = m - \frac{1}{m} \)[/tex], we get:
[tex]\[ x^2 = \left(m - \frac{1}{m}\right)^2 \][/tex]
Expanding the right-hand side:
[tex]\[ x^2 = m^2 - 2\cdot m \cdot \frac{1}{m} + \left(\frac{1}{m}\right)^2 \][/tex]
[tex]\[ x^2 = m^2 - 2 + \frac{1}{m^2} \][/tex]
Since we know that [tex]\( m^2 + \frac{1}{m^2} = 9 \)[/tex], we can substitute this value into the equation:
[tex]\[ x^2 = 9 - 2 \][/tex]
[tex]\[ x^2 = 7 \][/tex]
To find [tex]\( x \)[/tex], we take the square root of both sides:
[tex]\[ x = \sqrt{7} \][/tex]
Therefore, the value of [tex]\( m - \frac{1}{m} \)[/tex] is:
[tex]\[ \sqrt{7} \][/tex]
So, the answer is:
c. [tex]\( \sqrt{7} \)[/tex]
