Simplifying Expressions
Simplifying expressions mean rewriting the same
algebraic expression with no like terms and in a compact manner. To simplify
expressions, we combine all the like terms and solve all the given brackets, if
any, and then in the simplified expression, we will be only left with unlike
terms that cannot be reduced further. Let us learn more about simplifying
expressions in this article.
How to Simplify Expressions?
Before learning about simplifying expressions, let us
quickly go through the meaning of expressions in math. Expressions refer to
mathematical statements having a minimum of two terms containing either
numbers, variables, or both connected through an addition/subtraction operator
in between. The general rule to simplify expressions is PEMDAS - stands for
Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. In
this article, we will be focussing more on how to simplify algebraic expressions.
Let's begin!
We need to learn how to simplify expressions as it
allows us to work more efficiently with algebraic expressions and ease out our
calculations. To simplify algebraic expressions, follow the steps given below:
Step 1:
Solve parentheses by adding/subtracting like terms inside and by multiplying
the terms inside the brackets with the factor written outside.
For example, 2x(x + y) can be simplified as 2x2
+ 2xy.
Step 2:
Use the exponent rules to simplify terms containing exponents.
Step 3:
Add or subtract the like terms.
Step 4:
At last, write the expression obtained in the standard form (from highest power
to the lowest power).
Let us take an example for a
better understanding.
Simplify the expression: x(6 – x) – x(3 – x).
Here, there are two parentheses both having two unlike
terms. So, we will be solving the brackets first by multiplying x to the terms
written inside.
x(6 - x) can be simplified as 6x - x2,
-x(3 - x) can be simplified as -3x + x2.
Now, combining all the terms will result in 6x - x2
- 3x + x2. In this expression, 6x and -3x are like terms, and -x2
and x2 are like terms. So, adding these two pairs of like terms will
result in (6x - 3x) + (-x2 + x2). By simplifying it
further, we will get 3x, which will be the final answer. Therefore, x(6 – x) –
x(3 – x) = 3x.
Look at the image given below showing another
simplifying expression example.
Rules for Simplifying Algebraic
Expressions
The basic rule for simplifying expressions is to
combine like terms together and write unlike terms as it is. Some of the rules
for simplifying expressions are listed below:
To add two or more like terms, add their coefficients
and write the common variable with it.
Use the distributive property to open up brackets in
the expression which says that:
a(b + c) = ab +
ac.
If there is a negative sign just outside parentheses,
change the sign of all the terms written inside that bracket to simplify it.
If there is a 'plus' or a positive sign outside the
bracket, just remove the bracket and write the terms as it is, retaining their
original signs.
Simplifying Expressions with Exponents
To simplify expressions with exponents is done by
applying the rules of exponents on the terms. For example, (3x2)(2x) can be
simplified as 6x3. The exponent rules chart that can be used for simplifying
algebraic expressions is given below:
Example:
Simplify: 2ab + 4b(b2 - 2a).
To simplify this expression, let us first open the
bracket by multiplying 4b to both the terms written inside. This implies, 2ab +
4b(b2) - 4b(2a). By using the product rule of exponents, it can be
written as 2ab + 4b3 - 8ab, which is equal to 4b3 - 6ab.
This is how we can simplify expressions with exponents
using the rules of exponents.
Simplifying Expressions with
Distributive Property
Distributive property states
that an expression given in the form of:
x(y + z) can be simplified as xy + xz.
It can be very useful while simplifying expressions.
Look at the above examples, and see whether and how we have used this property
for the simplification of expressions.
Let us take another example of simplifying 5(2a + 4a +
3b) - 7b using the distributive property.
Remember : x(y + z) = xy + xz
5(2a + 4a + 3b) - 7b
= 5(6a + 3b) - 7b
= 5 × 6a + 5 × 3b - 7b
= 30a + 15b - 7b
= 30a + 8b
Therefore, 5(2a + 4a + 3b) - 7b is simplified as 30a +
8b.
Now, let us learn how to use the distributive property
to simplify expressions with fractions.
Simplifying Expressions with Fractions
When fractions are given in an expression, then we can use the distributive property and the exponent rules to simplify such expression.
All three are unlike terms, so it is the simplified
form of the given expression.
While simplifying expressions with fractions, we have
to make sure that the fractions should be in the simplest form and only unlike
terms should be present in the simplified expression.
