If $m+\frac{1}{m}=6$, find the value of $m^2+\frac{1}{m^2}$

The correct option is 34.

We are given the algebraic equation:

$m+\frac{1}{m}=6$

We need to find the value of:

$m^2+\frac{1}{m^2}$

To relate the given expression to the required expression, we can square both sides of the given equation using the algebraic identity ${(a+b)}^2=a^2+2ab+b^2$.

Squaring both sides of $m+\frac{1}{m}=6$ gives:

${(m+\frac{1}{m})}^2=6^2$

Expanding the left-hand side using the identity:

$m^2+2·m·\frac{1}{m}+{\left(\frac{1}{m}\right)}^2=36$

Simplifying the product term $m·\frac{1}{m}=1$:

$m^2+2+\frac{1}{m^2}=36$

Subtract 2 from both sides of the equation to solve for $m^2+\frac{1}{m^2}$:

$m^2+\frac{1}{m^2}=36-2$

$m^2+\frac{1}{m^2}=34$

Thus, the value of $m^2+\frac{1}{m^2}$ is 34.