An expression can easily be simplified by combining like terms and performing basic arithmetic operations on it. Parentheses ( ) and brackets [ ] can be used to group terms, and while simplifying an expression it is good to follow the order of operations.

For example, $\frac{x^{2}-1}{x+1}$ can be written as $\frac{x^{2}-1}{x+1}$ = $\frac{(x+1)(x-1)}{x+1}$. Strike out the common factor (x + 1) as it is the common term. Therefore the solution is (x - 1).

Algebraic expressions contains variables and also numbers. When an algebraic expression is simplified the resulting expression will be simpler than the original. There is no standard procedure for simplifying algebraic expressions as they are of different kinds.
However they can grouped into three types:

1. Easy to simplify.
2. Need some preparation while simplifying.
3. Cannot be simplified.

Solved Example

Question: Simplify : 7x + 8y + 2 + 4x - 7y + 5
The given expression can be easily simplified by grouping and combining like terms.
7x + 4x  when combined gives 11x,
8y - 7y  is y and,
2 + 5 is 7.
Therefore the resulting expression is 11x + y + 7.

A rational expression can be easily simplified by completely factoring numerator and denominator and thereafter reducing the common factors.

Solved Example

Question: Solve $\frac{14x - 7y}{2x-y}$
Step -1: The numerator (14x - 7y) can be written as 7(2x - y)

$\frac{14x - 7y}{2x-y}$$\frac{7(2x - y)}{2x-y}$

Step - 2: Cancel out the common factor (2x - y)

Step - 3: Therefore, $\frac{14x - 7y}{2x-y}$  = 7

When simplifying radicals it is necessary to know the two simple rules. They are:

Product rule:

If $\sqrt[n]{a}$ and $\sqrt[n]{b}$ are real numbers and n is a natural number then

$\sqrt[n]{a}$ $\times$ $\sqrt[n]{b}$ = $\sqrt[n]{ab}$

Quotient rule:

If $\sqrt[n]{a}$ and $\sqrt[n]{b}$ are real numbers, b $\neq$ 0 and n is a natural number then

$\frac{\sqrt[n]{a}}{\sqrt[n]{b}}$ = $\sqrt[n]{\frac{a}{b}}$

By following the above rules we can easily simplify a problem.

Solved Example

Question: Solve $\sqrt{\frac{7}{16}}$
Given $\sqrt{\frac{7}{16}}$

By using quotient rule this can be written as

$\sqrt{\frac{7}{16}}$ = $\frac{\sqrt{7}}{\sqrt{16}}$

$\Rightarrow$ $\frac{\sqrt{7}}{\sqrt{16}}$ = $\frac{\sqrt{7}}{4}$

The solution is $\frac{\sqrt{7}}{4}$

While simplifying logarithmic expressions it is necessary to remember its properties which aids in simplifying an expression. They are:

1. $log_{b}(mn)$ = $log_{b}(m)$ + $log_{b}(n)$
2. $log_{b}(\frac{m}{n})$ = $log_{b}(m)$ - $log_{b}(n)$
3. $log_{b}(m^{n})$ = n $\times log_{b}(m)$
4. $log_{b}x$ = $\frac{log_{a}x}{log_{a}b}$

Solved Example

Question: $log_{4}$ $(\frac{16}{x})$
By using the second property of logarithms the given problem can be written as

$log_{4}$$(\frac{16}{x})$ = $log_{4}(16)$ - $log_{4}(x)$    

But $log_{4}(16)$ = 2

Therefore $log_{4}$ $(\frac{16}{x})$  = 2 - $log_{4}(x)$

A complex expression will be in the form of a + ib where a and b are real numbers and i is the imaginary part. If there is no real part of an complex number then it is said to be purely imaginary where as if there is no complex part in a complex number then it is said to be a real number and $i^{2}$ = -1.

Solved Example

Question: Simplify: (6 - 2i) - (7 + 3i)
(6 - 2i) - (7 + 3i) = 6 - 2i - 7 - 3i
 $\Rightarrow$  (6 - 7) + (- 2i - 3i)
 $\Rightarrow$  (- 1 - 5i)
= - (1 + 5i)

A variable expression is a combination of numbers (or constants), operations, and variables. It is simplified by combining like terms and is found mostly in the process of simplifying an expression.
The following are the examples of variable expressions: 3x + 5y, $\frac{-9}{8}$x + 7, $\frac{-3}{4}$x + 5

Solved Example

Question: Simplify:10 (20x + 5) + 3
(200x + 50 ) + 3
$\Rightarrow$  200x + 53

To simplify expressions consisting of fractions consider the following steps:

1. Find the factors for numerator as well as denominator.
2. Now write down the common factors for numerator and denominator.
3. Considering the common factors divide both the top and bottom of the fraction by the Greatest Common Factor(GCF).

Solved Example

Question: $\frac{18xy^{3}z^{2}}{3xyz}$
$\frac{18xy^{3}z^{2}}{3xyz}$ =  $\frac{3 \times  6 \times x \times y \times y \times y \times z \times z}{3 \times x \times y\times z}$
= $6y^{2}z$

To simplify an expression with exponents, simplify the term using multiplication, division and power to power rules. Combine the like terms and arrange them. The variables with highest order of exponent should be put first.

Solved Example

Question: $15x^{2}$ + 7y - 2y + $\frac{x^{7}}{x^{3}}$ + 7
$15x^{2}$ + 7y - 2y + $\frac{x^{7}}{x^{3}}$ + 7 = $15x^{2}$ + 5y + $x^{4}$ + 7
Placing highest order exponent first we get,
$x^{4}$ + $15x^{2}$ + 5y +7

Boolean expression is an expression resulting in a boolean value where the value can be either true or false. Boolean expression makes the decision.
Roots are the opposite operation of applying exponents. While simplifying an expression containing square root take out anything that is a perfect square where the number are broken down into factors.

Example: Given $\sqrt{4500x^{2}y^{3}}$

$\sqrt{4500}$ = $\sqrt{45\times100}$ = $\sqrt{5\times9\times100}$ = $3\times10\sqrt{5}$

$\sqrt{x^{2}}$ = x

and $\sqrt{y^{3}}$ = $\sqrt{y^{2}}\times y$ = $y\sqrt{y}$

Therefore $\sqrt{4500x^{2}y^{3}}$ = 30$xy$$\sqrt{5y}$

Solved Examples

Question 1: $\frac{3x^{2}-3x}{3x^{3} -6x^{2} + 3x}$
For the given expression 3x is a common factor for both numerator and the denominator.

$\Rightarrow$ $\frac{3x^{2}-3x}{3x^{3} -6x^{2} + 3x}$ = $\frac{3x(x-1)}{3x(x^{2} -2x +1)}$

Cancel out the common term 3x. $(x^{2} -2x +1)$ is $(x-1)^{2}$ which is (x + 1)(x - 1)

So we have $\frac{x -1}{(x - 1)(x + 1)}$ = $\frac{1}{(x + 1)}$

Therefore the solution is $\frac{1}{(x + 1)}$

Question 2: Simplify 17y + $75x^{4}$ - 12y + $\frac{x^{5}}{x^{3}}$ + 7 + 3.
17y + $75x^{4}$ - 12y + $\frac{x^{5}}{x^{3}}$ + 7 + 3 = 5y + $75x^{4}$ + $x^{2}$ + 11

Placing highest order exponent first we get,

$75x^{4}$ + 5y + $x^{2}$ + 11.