Simplify the followings expressions:
\[\begin{align}
  & \left( i \right)45+\left( -12 \right)-12 \\
 & \left( ii \right)123+89-\left( -11 \right) \\
 & \left( iii \right)-85-15+200 \\
 & \left( iv \right)96+\left( -16 \right)-\left( -20 \right) \\
 & \left( v \right)36-\left( -14 \right)+25 \\
 & \left( vi \right)155+\left( -25 \right)+50 \\
 & \left( vii \right)23+\left( -56 \right)-\left( -56 \right) \\
 & \left( viii \right)45-85-15+\left( -25 \right) \\
 & \left( ix \right)78-39+\left( -19 \right)-\left( -29 \right) \\
 & \left( x \right)450-255+\left( -230 \right)+560 \\
\end{align}\]

Hint: In this question, we are given ten expressions and we need to find their values for this we will understand some basic rules of integers which are:
(i) Multiplication of two negative signs gives us a positive sign.
(ii) Multiplication of a positive sign with a negative sign gives us a negative sign.
(iii) For numbers in the form -a-b, we just have to add a and b and then put a negative sign in the front.
(iv) For numbers in the form a-b or -a+b, we just have to subtract the number and the sign will be according to the sign of the bigger number among a and b.

Complete step-by-step solution
Let us start by simplifying all expressions one by one.
$\left( i \right)45+\left( -12 \right)-12$.
Let us solve the bracket first, we know multiplication of a positive sign with a negative sign gives a negative sign, so we get $45-12-12$.
Now, 45-12 = 33 with the positive sign because 45 is a higher number having a positive sign, we get $33-12=21$. (Positive sign because 33 is higher than 12).
Hence $45+\left( -12 \right)-12=21$.
$\left( ii \right)123+89-\left( -11 \right)$.
Solving brackets first, we know multiplication of negative sign with negative sign gives us positive sign, so we get $123+89+11$.
Adding 123 and 89 we get: $212+11=223$.
Hence $123+89-\left( -11 \right)=223$.
$\left( iii \right)-85-15+200$.
Adding 85 and 15 gives us 100, since both are negative so 100 will also be negative. Hence we get: $-100+200=100$ (Positive sign because 200 is higher than 100).
Hence $-85-15+200=100$.
\[\left( iv \right)96+\left( -16 \right)-\left( -20 \right)\].
Multiplication of positive with negative sign gives negative sign whereas multiplication of negative sign with negative sign gives positive sign, so we get: $96-16+20$.
Subtracting 16 from 96 we get: $80+20=100$. (Positive sign with 80 because 96 is higher than 16).
Hence \[96+\left( -16 \right)-\left( -20 \right)=100\].
$\left( v \right)36-\left( -14 \right)+25$.
We know $\left( - \right)\left( - \right)=\left( + \right)$ so we get: $36+14+25$.
Adding 36 and 14 we get: $50+25=75$.
Hence, $36-\left( -14 \right)+25=75$.
\[\left( vi \right)155+\left( -25 \right)+50\].
We know $\left( + \right)\left( - \right)=\left( - \right)$ so we get: $155-25+50$.
Adding 130 and 50 we get: $130+50=180$. (Positive sign with 130 because 155>25).
Hence, \[155+\left( -25 \right)+50=180\].
$\left( vii \right)23+\left( -56 \right)-\left( -56 \right)$.
We know, $\left( + \right)\left( - \right)=\left( - \right)\text{ and }\left( - \right)\left( - \right)=\left( + \right)$ so we get: $23-56+56$.
We know $+a-a=0$ so we get: $23+0=23$.
Hence $23+\left( -56 \right)-\left( -56 \right)=23$.
$\left( viii \right)45-85-15+\left( -25 \right)$.
We know $\left( + \right)\left( - \right)=\left( - \right)$ so we get: $45-85-15-25$.
Subtracting 45 from 85 we get 40 so we get: $-40-15-25$. (Negative sign with 40 because 85>45).
Adding 40 and 15 we get 55 but putting a negative sign with 55 because both 40 and 15 are negative. $-55-25=-80$.
Hence, $45-85-15+\left( -25 \right)=-80$.
$\left( ix \right)78-39+\left( -19 \right)-\left( -29 \right)$.
We know $\left( + \right)\left( - \right)=\left( - \right)\text{ and }\left( - \right)\left( - \right)=\left( + \right)$ so we get: $78-39-19+29$.
Subtracting 39 from 78 we get $39-19+29$ (Positive sign with 39 because 78>39).
$20+29$ (Positive sign with 20 because 39>19) $49$.
Hence $78-39+\left( -19 \right)-\left( -29 \right)=49$.
$\left( x \right)450-255+\left( -230 \right)+560$.
We know that $\left( + \right)\left( - \right)=\left( - \right)$ so we get: $450-255-230+560$.
Subtracting 255 from 450 we get: $195-230+560$ (Positive sign with 195 because 450>255).
Subtracting 195 from 230 we get: $-35+560$ (Negative sign with 35 because 230>195).
$525$ (Positive sign with 525 because 560>35).
Hence $450-255+\left( -230 \right)+560=525$.

Note: Students should take care of positive and negative signs while solving expressions. Use the rules carefully at every step. Do not try to solve more than two numbers in one step as it can cause mistakes. While subtracting make sure to use signs of the greater number.