In this section we first explain the two possible types of simplifications that can be applied to an expression:

Then we give an exhaustive

Just as you should do, the Algebra Coach program always returns its results in simplest form. You can type an unsimplified expression into the Algebra Coach’s listbox and get it put into simplest form by clicking the Simplify button.

In general all integers, fractions, radicals and the numbers

An expression containing only exact numbers can actually be simplified in two different ways:

- to a result that is also expressed in exact form, or
- rounded to a result expressed in floating point form.

Thus before you begin simplifying an

The

Here is a checklist of various simplifications that you can apply to an expression. We have organized them into categories:

- sort the terms of an expression from highest degree to lowest
- sort the factors of a term into alphabetical, coefficient-first order
- drop zero terms, unnecessary + signs and unnecessary factors of 1
- drop unnecessary brackets

- distribute + and − signs
- add like terms
- combine several signs of factors into a single sign
- combine several numerical factors into a single factor
- combine exponential factors with the same base using the properties of exponents
- evaluate exponentials
- evaluate functions
- be aware of alternative notations used for square roots, exponential functions and reciprocals

- simplify a fraction multiplied or divided by a non-fraction
- replace the product of two fractions by a single fraction
- replace the quotient of two fractions by a single fraction
- reduce common fractions, fraction coefficients and algebraic fractions to lowest terms

- simplify exponentials with bases or exponents of 0 or 1
- simplify exponentials with negative exponents
- simplify exponentials whose bases are exponentials
- simplify exponentials whose bases are products or quotients
- simplify exponentials whose exponents are fractions

- use the fact that
- simplify the square root of a perfect square
- simplifications involving the square roots of integers
- simplification of square roots involving fractions

- simplify the logarithm of an exponential
- simplify the logarithm of a reciprocal
- simplify the ratio of two logarithms of numbers

- cancel a function composed with its inverse function
- simplify certain trigonometric function compositions

- simplify by eliminating csc, sec and cot in favor of sin, cos and tan
- simplify by eliminating negative arguments of sin, cos and tan
- simplify sin, cos and tan by removing π/2 or 90° phase shifts
- simplify using Pythagoras’ theorem

- express the square roots of negative numbers as imaginary numbers
- evaluate the sum, difference, product or quotient of two complex numbers
- use Euler’s formula to convert complex numbers between polar and rectangular coordinates
- evaluate functions whose input or output are complex numbers

A factor of 1 in a term does not have to be written since 1 is the multiplicative identity. Also a + sign in front of a single term or in front of the first term of a sum of terms is understood and does not have to be written. Thus these simplifications should be made: The only exception is when the 1 is used as a placeholder as in this example (otherwise the numerator would be blank): so no simplification is possible.

Now, if part of an expression is already at the top of the list then putting brackets around it is unnecessary and those brackets should be omitted. Some examples are:

- Be aware that in the last example
*c*is in the numerator, not the denominator. Since many people might be confused by this it might be a good idea to leave the brackets in, or even better, to write the fraction as .

- In some cases brackets do not appear in the typeset form but they are necessary when typing in the expression. They are deemed to be in certain locations such as the exponent and the numerator above and the denominator below a horizontal division line. For example: Click here for more information on invisible brackets.

The quantitya(b+c) =a · b + a · c

- adding a negative is the same thing as subtracting
(see the second and third examples below),

- when distributing a − sign over a sum you must flip the sign of each term of the sum (see the last two examples below).

before simplifyingafter simplifyinga+b+ (c+d)a + b + c + da+ (−b)a−b( a − b) + (−c + d)a − b − c + da− (−b + c)a + b−c− (− b + c)b−c

- In the second example the like terms happen to be in the numerator of the fraction.
- In the third example we are
*subtracting*instead of*adding*like terms and we write the result as*x*instead of 1*x*. - In the last example the coefficients happen to be fractions and . Click here to see how to add common fractions.

- The first and second examples both contain the floating point number, 7.2, so all the numbers are combined into the single floating point number, 5.76.
- The second example results in an algebraic fraction. The coefficient, 5.76, is put in the numerator (or possibly in front of the fraction).
- In the third example the numbers 7, 4 and 5 are all considered to be exact so the resulting coefficient is kept exact and written as the fraction 28/5.

If the argument of the function is an exact number then the value of the function is also an exact number. However only a few exact arguments have

- It shows that the following forms are all equivalent.
Notice that in all cases
*a*and*c*are in the numerator and*b*is in the denominator. The Algebra Coach simplifies all the other forms to the last form:

- It shows that multiplying by the reciprocal of
*b*is the same as dividing by*b*:

- By factoring the numerator and denominator and then reversing this simplification
we can reduce a fraction to its simplest equivalent fraction.
Here is an example:

This is a very useful result: The result of dividing two fractions with equal denominators is a fraction consisting of just the two numerators.

Here is another useful example: a non-fraction divided by a fraction. Notice how the fraction (because it comes from the denominator) is inverted:

- In a common fraction. Here is an example:
- In a coefficient that is a fraction. Here is an example:
- In an algebraic fraction. Here is an example:

- Anything raised to the power 0 equals 1.
Click here for more information.
*b*^{ 0}= 1 - Any “thing” raised to the power 1 just equals the “thing”:
*b*^{ 1}=*b* - 1 raised to any power equals 1.
1

^{ x}= 1 - 0 raised to a power can equal 0, 1 or be undefined. Click here for more information.

The above formula can be generalized to any negative exponent: Here are some examples: Click here for more information and examples.

Note that this simplification applies only to products or quotients. It does NOT apply to sums or differences! Click here for more information and examples.

Here is an example that makes use of this simplification: Click here for more information.

- Click here for more information.

- Click here for more information.

- If the radicand (the quantity under the square root) contains a
**perfect square factor**then remove the factor from the radicand. Here is an example: - Sometimes replacing the product of square roots of integers by the square root of a product will result in a perfect square factor which can then be removed as described in the first bullet. Click here for more information. Here is an example:

- Suppose that the denominator of a fraction contains a
*square*root. Then multiply both the numerator and denominator of the fraction by that square root and simplify. This will eliminate the radical from the denominator. Here is an example:

- Suppose that a square root contains a fraction.
Then multiply both the numerator and denominator of the fraction by the denominator of
the fraction and simplify. This will eliminate the radical from the denominator.
Here is an example:

- Suppose that the denominator of a fraction is a binomial,
*a*+*b*, and that one or both of the terms is a radical. Then multiplying the numerator and denominator of the fraction by the**binomial conjugate**,*a*−*b*, of the denominator and distributing will eliminate all radicals from the denominator. Click here for more information. Here is an example:

These inverse function compositions often occur when we solve an equation by doing the same thing to both sides of the equation. For example the equation:

10can be solved for^{ x}= 33

log (10According to the fourth row in the above table the left side of the equation then simplifies like this:^{ x }) = log (33)

and the right side can then be evaluated.x= log (33)

These function compositions often occur when we solve an equation by doing the same thing to both
sides of the equation. For example suppose that in the triangle shown to the right we want to find the tangent of the angle
θ without actually finding θ.
This can be done by noticing that
θ = arcsin (0.4)and taking the tangent of both sides of this equation. This gives: tan (θ) = tan (arcsin (0.4)) |

The following compositions of trig functions also simplify:

arccos(sin(θ)),They all simplify to π/2 radians − θ (or to 90° − θ in degree mode). To prove this notice that these 4 statements apply to the triangle shown: Now simply apply the operations indicated by the arrows:

arcsin(cos(θ)),

arctan(1/tan(θ)).

sin(−α) = −sin(α)

cos(−α) = cos(α)

tan(−α) = −tan(α)

- sin(
*x*+ π/2) = cos(*x*) or sin(*x*+ 90°) = cos(*x*)

- cos(
*x*+ π/2) = −sin(*x*) or cos(*x*+ 90°) = −sin(*x*))

- tan(
*x*+ π/2) = −1/tan(*x*) or tan(*x*+ 90°) = −1/tan(*x*)

sin(These identities are an important part of the algorithms used by the Algebra Coach and by calculators and computers to evaluate the sin, cos, and tan functions.x+ 270°) = cos(x+ 180°) = −sin(x+ 90°) = −cos(x).

sin^{2}(α) + cos^{2}(α) = 1

1 − sin^{2}(α) = cos^{2}(α)

1 − cos^{2}(α) = sin^{2}(α)

An important special case is Euler’s identity,

Click here for more information.e^{i π}= −1.

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