Simplifying Expressions

The combination of constants, variables with exponents, and mathematical operators like addition, subtraction, multiplication, and division is called an algebraic expression. Some examples of algebraic expressions are:

• \(3x + 2\)
• \(x^2 - 5x + 6\)
• \(a + b - 7\)

Making an algebraic expression easier to read and work with, without changing its value, is known as simplifying expressions. It is a fundamental skill in algebra that helps in solving equations, graphing functions, and understanding mathematical relationships. We can simplify expressions by:

1. Combining like terms - The terms that have the same variables, like 2x and 7x, can be added or subtracted. 

2. Removing brackets - We can remove parentheses by using rules such as the distributive property.

3. Rewriting an expression - Rewriting an expression through simplification makes it easier by combining like terms and reducing it to its simplest form. 

Example: Simplify \(3x + 2x + 5\)

Combining like terms: \((3×x + 2×x) + 5 = 5x + 5\).

\(5x + 5 \) is the simplified expression.

The basic rule of simplifying an expression is to combine like terms and leave the unlike terms unchanged.  We try to make the expression shorter and easier by the following rules.

Rule 1: Add like terms - If two or more terms have the same variable, just add their coefficients. For example, \(2x + 5x = 7x\).

Rule 2:  Use the distributive property - If there is any number outside the brackets, multiply the number by everything inside the brackets. Example: \(2(x + 3) = 2x + 6.\)

Rule 3: Minus sign before brackets - If there is a minus sign before a bracket, change the signs of everything inside the bracket. If the given equation is like -(x + 2), we have to change the sign of everything inside the bracket, and it becomes\( -1 × (x + 2) = -x - 2\).

Rule 4: Plus sign before brackets - If there is a plus sign before the brackets, removing the brackets does not change the signs of the terms inside. Example: \( -1 × (x + 2) = -x - 2\).

The method which is used for simplifying expressions is the FOIL method. The FOIL method is used to multiply two binomials. Binomials are expressions that have two terms, for example, \((x + a)(x + b)\). FOIL stands for,

F - First

O - Outer

I - Inner

L - Last

We can multiply in the FOIL order to make sure that we are multiplying all the terms. Let’s go through the steps of the FOIL method using an example.

Step 1: First, in the binomials, multiply the first terms. 
Example: \((x + 1)(x + 4)\)
\(x × x = x²\)

Step 2: Outer, multiply the outer terms. Like, multiply the first term in the first binomial with the second term in the second binomial. 
\( x × 4 = 4x\)

Step 3: Inner, multiply the inner terms of the binomials. The second term from the first binomial with the first term in the second binomial.
\(1 × x = x\)

Step 4: Last, multiply the last terms of each binomial. Multiply the second terms of both the given binomials.
\(1 × 4 = 4\)

Step 5: Add all the terms, combine them, and you will get the final answer.
\(x^2 + 4x + x + 4 = x^2 + 5x + 4.\)

When we simplify expressions with exponents, we use special rules to make them easier. The rules for simplifying expressions are:

1. Any non-zero base raised to the power of 0 is equal to 1:
a⁰ = 1, where a ≠ 0.

2. Any number or variable raised to the power of 1 remains unchanged. For example, a¹ = a.

3. When multiplying the same base, add the exponents:
\(x^2 × x^3 = x^5\).

4. When dividing with the same base, subtract the exponents: \(x^5 ÷ x^3 = x^2\).

5. If there is a negative exponent, we have to flip it: \(x^{-2} = 1/x^2\).

6. If we are multiplying a power with another power, multiply the exponents:\( (x^2)^3 = x^6\).

Example: Simplify \(3a + 2a(2a)\).
Multiply \(2a × 2a = 4a^2\)
Now add \(3a + 4a^2\).

Here, we cannot combine both the terms because a and a² are unlike terms.

The distributive property allows you to multiply a number by each term inside the brackets. If the number is in the form of \(a(b + c)\), it can be simplified as \(ab + bc\). Let us look at an example,
Simplify \(3(a + b + 4)\)

1. Using the distributive property, multiply the number 3 by the terms inside the brackets.

\(3 × a = 3a\)

\(3 × b = 3b\)

\(3 × 4 = 12\)

Combine all the terms to get the final answer.

Therefore, the final answer is: \(3a + 3b + 12\).

When simplifying expressions with fractions, we still use the distributive property and some fraction rules to simplify them. Given below are the steps for simplifying expressions with fractions,

Step 1: Use the distributive property to remove the brackets.

Step 2: When multiplying fractions, multiply the numerators together and the denominators together.

Step 3: Simplify the fractions if we can.

Step 4: If the terms are unlike, just write them like that; you can’t combine them.

Example: Simplify \(\frac{1}{3}x + \frac{2}{5}(5x + 10) \)

Step 1: Use the distributive property.

Multiply 25 the terms inside the brackets.

\(\frac{2}{5}(5x + 10) \)

\(\frac{2}{5} \times 5x = 2x \)

\(\frac{2}{5} \times 10 = 4 \)
So now we have \(13x + 2x + 4 \)

Step 2: Check for like terms.

Leave the expression as it is because we cannot combine them. So the final answer is:
\(13x + 2x + 4 \)
 

Simplifying expressions helps make algebra easier by combining like terms and solving brackets. Regular practice and careful attention to rules improve accuracy and speed.

• Always look for like terms first and combine them carefully.
   
• Solve all brackets and parentheses before combining terms.
   
• Apply the distributive property correctly to eliminate multiplication across brackets.
   
• Keep an eye on positive and negative signs to avoid mistakes.
   
• Practice with different types of expressions regularly to build speed and accuracy.

Simplifying expressions helps solve everyday problems efficiently, from budgeting to cooking. It makes calculations quicker and easier to understand.

• Budgeting and finance: Simplifying expressions helps calculate total expenses or income by combining similar costs or earnings.
   
• Shopping discounts: When applying multiple discounts or offers, simplifying expressions makes it easier to find the final price.
   
• Construction projects: Engineers simplify measurements and materials calculations to estimate total resources needed.
   
• Cooking and recipes: Simplifying ingredient quantities when scaling recipes up or down ensures accurate proportions.
   
• Travel planning: Calculating total distance, time, or fuel consumption becomes easier by simplifying expressions involving multiple routes or speeds.