What is the difference between $-{{5}^{2}}$ and ${{\left( -5 \right)}^{2}}$?

Hint: To obtain the difference between the two values we can find the simplified value of both and draw a conclusion. Firstly we will solve the value where the negative sign is outside the bracket. Then we will solve the value whose negative sign is inside the value. Then finally we will check what is the difference between the two values obtained and draw our conclusion.

Complete step by step solution:
The values given are:
$-{{5}^{2}}$ and ${{\left( -5 \right)}^{2}}$
Firstly we will solve first value as below:
$-{{5}^{2}}=-25$…….$\left( 1 \right)$
As negative signs don’t have square power it will remain negative.
Next we will solve second value as below:
${{\left( -5 \right)}^{2}}={{\left( -1 \right)}^{2}}\times {{\left( 5 \right)}^{2}}$
$\therefore {{\left( -5 \right)}^{2}}=25$……$\left( 2 \right)$
As negative signs have power 2 and we know two negative forms a positive.
So from equation (1) and (2) we can see there is a difference of a negative sign only.

Hence the difference is that the $''-''$ sign is in brackets in one value and outside the bracket in another.

Note: If a negative sign is outside a bracket we multiply it inside the bracket once we have simplified value in it but when a negative sign is inside a bracket with some power we use the minus-minus plus concept. We have to keep an eye on which part of the expression is raised to power and which is not. This is a common mistake while solving such types of questions and that is why it is necessary that we know the difference between the two values.