Free Algebra
Tutorials!
Home
Miscellaneous Equations
Operations with Fractions
Undefined Rational Expressions
Inequalities
Writing Equations for Lines Using Sequences
Intersections of Lines and Conics
Graphing Linear Equations
Solving Equations with Log Terms and Other Terms
Quadratic Expresions - Complete Squares
Adding and Subtracting Fractions with Like Denominators
Multiplying a Fraction by a Whole Number
Solving Equations with Log Terms and Other Terms
Solving Quadratic Equations by Factoring
Locating the Solutions of the Quadratic Equation
Properties of Exponents
Solving Equations with Log Terms on Each Side
http:
Graphs of Trigonometric Functions
Estimating Products and Quotients of Mixed Numbers
Inequalities
The circle
Adding Polynomials
Adding Fractions with Unlike Denominators
Factoring Polynomials
Linear Equations
Powers of Ten
Straight Lines
Dividing With Fractions
Multiplication Property of Equality
Rationalizing Denominators
Multiplying And Dividing Fractions
Distance Between Points on a Number Line
Solving Proportions Using Cross Multiplication
Using the Quadratic Formula
Scientific Notation
Imaginary Numbers
Values of Symbols for Which Fractions are Undefined
Graphing Equations in Three Variables
Writing Fractions as Decimals
Solving an Equation with Two Radical Terms
Solving Linear Systems of Equations by Elimination
Factoring Trinomials
Positive Rational Exponents
Adding and Subtracting Fractions
Negative Integer Exponents
Rise and Run
Brackets
Multiplying Square Roots
Multiplying Polynomials
Solving Systems of Linear Inequalities
Multiplication Property of Radicals
A Quadratic within a Quadratic
Graphing a Linear Equation
Calculations with Hundreds and Thousands
Multiplication Property of Square and Cube  Roots
Solving Equations with One Log Term
The Cartesian Coordinate Plane
Equivalent Fractions
Adding and Subtracting Square Roots
Solving Systems of Equations
Exponent Laws
Solving Quadratic Equations
Factoring Trinomials
Solving a System of Three Linear Equations by Elimination
Factoring Expressions
Adding and Subtracting Fractions
The parabola
Computations with Scientific Notation
Quadratic Equations
Finding the Greatest Common Factor
Introduction to Fractions
Simplifying Radical Expressions Containing One Term
Polynomial Equations
Graphing and Intercepts
The Number Line
Adding and Subtracting Rational Expressions with Different Denominators
Scientific Notation vs Standard Notation
Powers
Factoring by Grouping
Extraneous Roots
Variables and Expressions
Linera Equations
Integers and Substitutions
Squares and Square Roots
Adding and Subtracting Rational Expressions with Different Denominators
Solving Linear Inequalities
Expansion of a Product of Binomials
Powers and Exponents
Finding The Greatest Common Factor
Quadratic Functions
The Intercepts of a Parabola
Solving Equations Containing Rational Expressions
http:
Subtracting Polynomials
Solving Equations
Adding Fractions with Unlike Denominators
Solving Systems of Equations by Substitution
Solving Equations
Product and Quotient of Functions
Try the Free Math Solver or Scroll down to Tutorials!

 

 

 

 

 

 

 

 
 
 
 
 
 
 
 
 

 

 

 
 
 
 
 
 
 
 
 

Please use this form if you would like
to have this math solver on your website,
free of charge.


Removing brackets 1

In order to simplify mathematical expressions it is frequently necessary to remove brackets. This means to rewrite an expression which includes bracketed terms in an equivalent form, but without any brackets. This operation must be carried out according to certain rules which are described in this leaflet.

1. The associativity and commutativity of multiplication

Multiplication is said to be a commutative operation. This means, for example, that 4×5 has the same value as 5×4. Eitherway the result is 20. In symbols, xy is the same as yx, and so we can interchange the order as we wish. Multiplication is also an associative operation. This means that when we want to multiply three numbers together such as 4×3×5 it doesn matter whether we evaluate 4×3 first and then multiply by 5, or evaluate 3×5 first and then multiply by 4. That is (4×3)×5 is the same as 4×(3×5) where we have used brackets to indicate which terms are multiplied first. Eitherway, the result is the same, 60. In symbols,we have (x × y) × z is the same as x × (y × z) and since the result is the same eitherway, the brackets make no difference at all and we can write simply x × y × z or simply xyz. When mixing numbers and symbols we usually write the numbers first. So

Example

Remove the brackets from

a) 4(2x)

b) a(5b)

Solution

a) 4(2x) means 4×(2 × x). Because of associativity of multiplication the brackets are unnecessary and we can write 4×2 × x

b) a(5b) means a ×(5b). Because of commutativity this is the same as (5b) × a that is (5 × b) × a. Because of associativity the brackets are unnecessary and we write simply 5 × b × a which equals 5ba. Note that this is also equal to 5ab because of commutativity.

Exercises

1. Simplify

Answers

2. Expressions of the form a( b + c ) and a( b - c )

Study the expression 4×(2 + 3).

By working out the bracketed term first we obtain 4×5 which equals 20. Note that this is the same as multiplying both the 2 and 3 separately by 4, and then adding the results. That is 4×(2 + 3) = 4×2 + 4×3 = 8 + 12 = 20.

Note the way in which the "4" multiplies both the bracketed numbers, "2" and "3". We say that the "4" distributes itself over both the added terms in the brackets - multiplication is distributive over addition.

Now study the expression 6×(8 - 3).

By working out the bracketed term first we obtain 6×5 which equals 30. Note that this is the same as multiplying both the 8 and the 3 by 6 before carrying out the subtraction:

6×(8 - 3) = 6×8 -6×3 = 48 - 18 = 30.

Note the way in which the "6" multiplies both the bracketed numbers. We say that the "6" distributes itself over both the terms in the brackets - multiplication is distributive over subtraction. Exactly the same property holds when we deal with symbols.

a (b + c) = ab + ac

a (b - c) = ab - bc

Example

Exercises

Remove the brackets from each of the following expressions simplifying your answers where appropriate.

Answers

All Right Reserved. Copyright 2005-2017