Question

Simplify the following:
$3a-\left[ a+b-\left\{ a+b+c-\left( a+b+c+d \right) \right\} \right]$

Answer 1

Hint: To simplify the above algebraic expression, we are going to first of all open the bracket in which $\left( a+b+c+d \right)$ is written. We know that when the negative sign multiplies with the positive sign then we get the negative sign as a result. This property, we will use when we open the brackets. Then we will open the curly brackets and do the algebraic addition or subtraction. Finally, we will open the hard brackets.

Complete step-by-step solution:
In the above problem, we are asked to simplify the following expression:
$3a-\left[ a+b-\left\{ a+b+c-\left( a+b+c+d \right) \right\} \right]$
To simplify it, we are first of all going to remove the open brackets in which $\left( a+b+c+d \right)$ is written and when we remove it we are going to use the property which states that when we multiply a positive sign with a negative sign then we will get the negative sign.
$3a-\left[ a+b-\left\{ a+b+c-a-b-c-d \right\} \right]$
Now, we are going to solve the expression written in the curly braces in which a, b and c will get cancelled out and we get,
$3a-\left[ a+b-\left\{ -d \right\} \right]$
After that, we are going to remove the curly braces and we know that when we multiply the negative sign with the negative sign then we get a positive sign and it will look as follows:
$3a-\left[ a+b+d \right]$
Now, we are going to remove the hard brackets and we get,
$\Rightarrow 3a-a-b-d$
Solving the above algebraic subtraction by taking $''a''$ as common from the first two terms we get,
$\begin{align}
  & a\left( 3-1 \right)-b-d \\
 & =2a-b-d \\
\end{align}$
Hence, we have simplified the given expression to $2a-b-d$.

Note: The trick to solving such problems in which so many brackets are given then start with the bracket which lies in the deepest part of the expression and start opening the bracket from there and then move forward. Also, you need to know how signs will change when we multiply two different signs or the same signs.