How to Perform Associative Operations

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How to Perform Associative OperationsBy Mark Zegarellifrom Basic Math and Pre-Algebra For Dummies

Addition and multiplication are both associative operations, which means that you can groupthem differently without changing the result. This property of addition and multiplication is also

called the associative property. Heres an example of how addition is associative. Suppose you

want to add 3 + 6 + 2. You can solve this problem in two ways:

In the first case, you start by adding 3 + 6 and then add 2. In the second case, you start by adding

6 + 2 and then add 3. Either way, the sum is 11.

And heres an example of how multiplication is associative. Suppose you want to multiply 5 2 4.You can solve this problem in two ways:

In the first case, you start by multiplying 5 2 and then multiply by 4. In the second case, you start

by multiplying 2 4 and then multiply by 5. Either way, the product is 40. In contrast, subtractionand division are nonassociative operations. This means that grouping them in different ways

changes the result.

Dont confuse the commutative property with the associative property. The commutative

property tells you that its okay to switch around two numbers that youre adding or multiplying.

The associative property tells you that its okay to regroup three numbers using parentheses.

Taken together, the commutative and associative properties allow you to completely rearrange

and regroup a string of numbers that youre adding or multiplying without changing the result.

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How to Solve Equations with Parentheses

By Mark Zegarelli from Basic Math and Pre-Algebra For Dummies

In math, parentheses ( ) are often used to group together parts of an expression. This helps

you to find the order of precedence when you work with equations. When it comes to evaluatingexpressions that contain parentheses, you can follow these steps:

1. Evaluate the contents of parentheses, from the inside out.

2. Evaluate the rest of the expression.

Big Four expressions with parentheses

Similarly, suppose you want to evaluate (1 + 15 5) + (3 6) 5. This expression contains two

sets of parentheses, so evaluate these from left to right. Notice that the first set of parentheses

contains a mixed-operator expression, so evaluate this in two steps starting with the division:

= (1 + 3) + (3 6) 5

= 4 + (3 6) 5

Now evaluate the contents of the second set of parentheses:

= 4 + 3 5

Now you have a mixed-operator expression, so evaluate the multiplication (3 5) first:

= 4 + 15

Finally, evaluate the addition:

= 11

So (1 + 15 5) + (3 6) 5 = 11.

Expressions with exponents and parentheses

As another example, try this out:

1 + (3 62 9) 22

Start out by working only with whats inside the parentheses. The first thing to evaluate there is

the exponent, 62:

= 1 + (3 36 9) 22

Continue working inside the parentheses by evaluating the division 36 9:

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= 1 + (3 4) 22

Now you can get rid of the parentheses altogether:

= 1 + 1 22

At this point, whats left is an expression with an exponent. This expression takes three steps,

starting with the exponent:

= 1 + 1 4

= 1 + 4

= 3

So 1 + (3 62 9) 22 = 3.

Expressions with parentheses raised to an exponent

Sometimes, the entire contents of a set of parentheses are raised to an exponent. In this case,evaluate the contents of the parentheses before evaluating the exponent, as usual. Heres an

example:

(7 5)3

First, evaluate 7 5:

= 23

With the parentheses removed, youre ready to evaluate the exponent:

= 8

Once in a rare while, the exponent itself contains parentheses. As always, evaluate whats in the

parentheses first. For example,

21(19 + 3 6)

This time, the smaller expression inside the parentheses is a mixed-operator expression. The partthat you need to evaluate first is underlined:

= 21(19 + 18)

Now you can finish off whats inside the parentheses:

= 211

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At this point, all thats left is a very simple exponent:

= 21

So 21(19 + 3 6) = 21.

Technically, you dont need to put parentheses around the exponent. If you see an expression in

the exponent, treat it as though it had parentheses around it. In other words, 2119 + 3 6 means the

same thing as 21(19 + 3 6)

.

Expressions with nested parentheses

Occasionally, an expression has nested parentheses: one or more sets of parentheses inside

another set. Here is the rule for handling nested parentheses:

When evaluating an expression with nested parentheses, evaluate whats inside the innermost setof parentheses first and work your way toward the outermost parentheses.

For example, suppose you want to evaluate the following expression:

2 + (9 (7 3))

The contents of the innermost set of parentheses are underlined, so evaluate these contents first:

= 2 + (9 4)

Next, evaluate whats inside the remaining set of parentheses:

= 2 + 5

Now you can finish things off easily:

= 7

So 2 + (9 (7 3)) = 7

As a final example, heres an expression that requires everything from this chapter:

4 + (7 (2(5 1) 4 6))

This expression is about as complicated as youre ever likely to see in pre-algebra: one set of

parentheses containing another set, which contains a third set. To start off, evaluate theunderlined part of the equation, which is in the third set of parentheses:

= 4 + (7 (24 4 6 ))

Now, whats left is one set of parentheses inside another set. Again, work from the inside out.The smaller expression here is 24 4 6, so evaluate the exponent first, then the multiplication,

and finally the subtraction:

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= 4 + (7 (16 4 6))

= 4 + (7 (16 24))

= 4 + (7 8)

Only one more set of parentheses to go:

= 4 + 56

At this point, finishing up is easy:

= 60

Therefore, 4 + (7 (2(5 1) 4 6)) = 60.

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Evaluate Mathematical Expressions

By Mark Zegarelli from Basic Math and Pre-Algebra For Dummies

The ideas of equality, expressions, and evaluation are vital concepts when working with

equations.An equation is a mathematical statement that tells you that two things have the samevalue in other words, its a statement with an equal sign. The equation is one of the most

important concepts in mathematics because it allows you to boil down a bunch of complicatedinformation into a single number.

Mathematic equations come in lots of varieties, with the two most common being arithmetic and

algebraic equations.

Three properties of equality

The three properties of equality are called reflexivity, symmetry, and transitivity:

Reflexivity says that everything is equal to itself. For example,

1 = 1

23 = 23

1,000,007 = 1,000,007

Symmetry says that you can switch around the order in which things are equal. For

example,

4 5 = 20, so 20 = 4 5

Transitivity says that if something is equal to two other things, then those two other

things are equal to each other. For example,

3 + 1 = 4 and 4 = 2 2, so 3 + 1 = 2 2

Because equality has all three of these properties, mathematicians call equality an equivalencerelation.

Here are a few examples of simple arithmetic equations:

2 + 2 = 4

3 4 = 12

20 2 = 10

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And here are a few examples of more-complicated arithmetic equations:

1,000 1 1 1 = 997

(1 1) + (2 2) = 5

What is an expression?

An expression is any string of mathematical symbols that can be placed on one side of an

equation. Here are a few examples of simple expressions:

2 + 2

17 + (1)

14 7

And here are a few examples of more-complicated expressions:

(88 23) 13

100 + 2 3 17

How to evaluate expressions

At the root of the word evaluation is the word value. In other words, when you evaluate

something, you find its value. Evaluating an expression is also referred to as simplifying,solving,orfinding the value ofan expression. The words may change, but the idea is the same boiling

a string of numbers and math symbols down to a single number.

When you evaluate an arithmetic expression, you simplify it to a single numerical value that

is, you find the number that its equal to. For example, evaluate the following arithmeticexpression:

7 5

How? Simplify it to a single number:

35

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How do you connect equality, expressions, and evaluation?

The Three Es equality, expressions, and evaluation are all connected. Evaluation allows

you to take an expression containing more than one number and reduce it down to a single

number. Then, you can make an equation, using an equal sign to connect the expression and thenumber. For example, heres an expression containing four numbers:

1 + 2 + 3 + 4

When you evaluate it, you reduce it down to a single number:

10

And now, you can make an equation by connecting the expression and the number with an equalsign:

1 + 2 + 3 + 4 = 10

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Find the Square Root of a Number

By Mark Zegarelli from Basic Math and Pre-Algebra For Dummies

A square root is the most common root operation. A root is the inverse operation of an exponent

(which means that it undoes an exponential operation), and so asquare rootis an operation thatundoes an exponent of 2. For example,

You can read the symbol either as the square root of or as radical. So, read

as either the square root of 9 or radical 9.

As you can see, when you take the square root of any square number, the result is the number

that you multiplied by itself to get that square number in the first place. For example, to find

you ask the question, What number when multiplied by itself equals 100? The answer in thiscase is 10, because

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Two Effective Ways to Solve Differential

Equations

By Steven Holzner

Part of the Differential Equations Workbook For Dummies Cheat Sheet

You can solve a differential equation in a number of ways. The two most effective techniques

you can use are the method of undetermined coefficients and the power series method.

The method of undetermined coefficients is a useful way to solve differential equations. Toapply this method, simply plug a solution that uses unknown constant coefficients into the

differential equation and then solve for those coefficients by using the specified initial

conditions.

Power series are another tool in your differential equation solving toolkit. You can substitute apower series such as the following into a differential equation:

Then all you have to do is find a recurrence relation that gives you the coefficient an.

How to Tell One Differential Equation from

AnotherBy Steven Holzner

Part of the Differential Equations Workbook For Dummies Cheat Sheet

Before you can solve a differential equation, you need to know what kind it is. There are severaldifferent types of equations, including linear, separable, exact, homogeneous, and

nonhomogeneous.

Linear differential equations deal solely with derivatives to the first power (forget about

derivatives raised to any higher power).

The power referred to here is the power the derivative is raised to, not the order of the derivative.

Heres a pretty typical-looking linear differential equation:

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Separable differential equations can be written so that all terms inx and all terms iny appear onopposite sides of the equation, as you can see in this example:

which can also be written as

Exact differential equations are those where you can find a function whose partial derivativescorrespond to the terms in the differential equation. Heres an example:

Homogeneous differential equations contain only derivatives ofy and terms involvingy. As you

can see in this equation, theyre also set to 0:

Nonhomogeneous differential equations are the same as homogeneous differential equations but

with one exception: They can only have terms involvingx and/or constants on the right side.

Heres an example of a nonhomogeneous differential equation:

The general solution of this nonhomogeneous differential equation:

is

where c1y1(x) + c2y2(x) is the general solution of the corresponding homogeneous differential

equation

andyp(x) is a particular solution to the nonhomogeneous equation.

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Solving Differential Equations Using Laplace

Transform Solutions

By Steven Holzner

Part of the Differential Equations Workbook For Dummies Cheat Sheet

Laplace transforms are a type of integral transform that are great for making unruly differentialequations more manageable. Simply take the Laplace transform of the differential equation in

question, solve that equation algebraically, and try to find the inverse transform. Heres the

Laplace transform of the functionf(t):

Check out this handy table of Laplace transforms for common functions whenever you dont

want to take the time to calculate a Laplace transform on your own.

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