### Evaluating equations

After you create an expression or equation, you can transform it by changing its mathematical repre­sentation. For example, you can multiply polynomials, factor terms, simplify expressions, and combine fractions.

 Original selection Expanded Multiplied out Multiplied out again

You can also evaluate equations by substituting values and performing computations.

 Original selection Expanded Value substituted Result computed

#### Transform or evaluate an expression

1)Select the expression, or part of it.

2)Choose a command on the Matrices or Functions page of the Equations pod. For example, select an expression and then choose Multiply Out from the Multiplication pop-up menu on the Functions page.

3)To transform or evaluate an expression, use the commands on the Matrix Commands pop-up menu of the Matrices page and the commands on the six pop-up menus of the Functions page.

Matrix Transpose

Transposes the rows and columns of a selected matrix.

Selected matrix

After transposing

Matrix Algebra

Performs matrix multiplication and addition, and evaluates dot and cross products in a selected matrix.

Selected matrix (or matrices)

After matrix algebra

After performing matrix algebra, you can use the Simplify command to combine like terms.

Adds selected fractions, expressing the result as a single fraction. This command also combines like terms.

Selected fraction

If the selected expression contains the sum of more than two fractions, Add Fractions adds only the first two. Use this command repeatedly to combine all the fractions.

Order Sum

Arranges polynomials in decreasing powers of a selected variable.

Before ordering

After ordering

Order Sum Reverse

Arranges polynomials in increasing powers of a selected variable. In the following example, you select an x—any x—before choosing Order Sum Reverse.

Before ordering

After ordering

Use the Multiplication pop-up menu on the Functions page to factor terms, multiply out expres­sions, and distribute multiplication and division over addition.

Factor

Factors a selected term out of a product.

Before factoring

After factoring

Factor Some

Factors a selected term from only those terms that contain it.

Before factoring

After factoring

Multiply Out

Simplifies the selected expression, distributes multiplication over addition, and expands a selected term raised to a positive integral power.

Selected expression

After multiplying

The Simplify command also simplifies the selected expression.

Multiply Out Once

Multiplies the first pair of factors on the left side of a selected expression.

Selected expression

After multiplying

If you use this command several times, the results appear in a different form.

Distribute

Performs a variety of operations depending on the expression you select, as shown in the following examples:

Selected expression

After distributing

Use the Add Fractions command to perform the reverse operation.

Selected expression

After distributing

Use the Factor command to perform the reverse operation.

Transforms products and quotients involving radicals and exponents into a single expression raised to a power.

Selected expression

After distributing

Use the Simplify command to perform the reverse operation.

Distribute Over Equality

Performs the same operation on both sides of an equal sign.

Selected expression

After distributing

Long Division

Performs long division in a fraction that contains a polynomial numerator and denominator.

In the following example, you select an x—any x—before choosing Long Division. You select an x in the fraction of the first result and choose Long Division again to obtain the second result.

 Before dividing Divided once Divided twice

Remove Division

Converts division to multiplication in a selected expression by changing positive exponents in a denominator to negative and negative exponents in a denominator to positive.

Selected expression

After removing division

To convert negative exponents to division operators, use the Remove Negative Powers command.

Remove Division 1 Level

Converts division to multiplication, as Remove Division does, but applies only to the first level of operators.

Selected expression

After removing division

To convert the first level of negative exponents into division operators, use the Remove Nega­tive Powers 1 Level command.

Remove Negative Powers

Converts negative exponents to positive in a selected expression by replacing multiplication with division and division with multiplication.

Selected expression

After removing negative powers

Remove Negative Powers 1 Level

Converts negative exponents to positive, as Remove Negative Powers does, but applies only to the first level. If the selected expression has no negative exponents at the first level, this command has no effect.

Selected expression

After removing negative powers

(no effect)

Number Crunch

Changes integers to floating-point numbers in a selected expression and then evaluates the expression.

Selected expression

After number crunch

Use the Show All Digits command to show up to 15 decimal places in a floating-point number.

Number Crunch displays NaN (Not a Number) for operations that result in undefined values. It displays Infinity for operations that result in a value too large to calculate. Small numbers can sometimes be evaluated to zero.

Selected expression

After number crunch

Show All Digits

Shows the full internal accuracy of a floating-point number.

Selected number

After showing all digits

Evaluate

Evaluates the selected expression. The operations performed depend on the type of expression selected, as shown in the following examples:

Rewrites an expression raised to a power less than 20 as a product.

Selected expression

After evaluating

Computes the factorial of an integer. If the result is too large to calculate, the result appears as Infinity.

Selected expression

After evaluating

Computes the determinant of a 1-by-1 or 2-by-2 matrix.

Selected expression

After evaluating

Extracts a term from a sum or product.

Selected expression

After evaluating

Evaluates a substitution just as the Evaluate Substitution command does.

Rewrites an Evaluate Between operator as a difference.

Selected expression

After evaluating

Rewrites the logarithm of a product as a sum of logarithms and rewrites a logarithm of a power as a product.

Selected expression

After evaluating

Rewrites a logarithm to a base in terms of natural logarithms.

Selected expression

After evaluating

Rewrites a choice function in terms of factorials.

Selected expression

After evaluating

Evaluate Substitution

Performs a specified substitution and then simplifies the expression. When an Evaluate Between operator is selected, this command rewrites the expression as a difference.

Selected expression

After evaluating

Evaluate Integrals

Rewrites a selected integral with a polynomial integrand. This command converts a definite integral into an indefinite integral evaluated between the upper and lower bounds. Use Eval­uate Integrals several times to integrate a polynomial fully.

 Selected expression Evaluated once Evaluated twice

Evaluate Derivatives

Evaluates a selected derivative, applying the chain rule to complex expressions when necessary. This command evaluates a partial derivative only when all dependencies are explicitly written.

This command treats elements in a total derivative as if they depended on the variable of differ­entiation. Use the Evaluate command to expand an nth derivative before using Evaluate Deriv­atives to evaluate it.

Before

After

Evaluate Derivatives 1 Level

Evaluates only the first level of a selected derivative.

Before

After

Use the Rules pop-up menu on the Functions page to store and apply rules that transform expres­sions.

For example, use Enter Rule to store a rule such as a = a + 3. Then use Apply Rule to substitute a + 3 for a in another expression.

Enter Rule

Stores a rule for substituting one expression for another. Select the expression you want to store as a rule before you choose Enter Rule. You can store only one rule at a time.

Designate Dummy

Designates a selected term to be a dummy variable. Use a dummy variable in a rule to substitute the expression on the right side of the rule for any variable, not just the variable explicitly named on the left. A dummy variable appears in boldface. For example, if you designate x in the rule x = a 2– as a dummy variable, you can substitute a 2– or x for any variable in an expres­sion.

Select the term you want to be a dummy variable before you choose Designate Dummy. Then use the Apply Rule command to replace a selected expression with the expression assigned to the dummy variable.

Apply Rule

Substitutes one term or set of terms for another in a selected expression, using the rule stored with the Enter Rule command.

 Rule entered Selected expression After rule is applied

Before you apply a rule that contains a dummy variable, select the term you want the variable to replace. In the following example, x is the dummy variable in the rule.

 Rule entered Before rule is applied After rule is applied

Simplify

Simplifies the selected expression. To evaluate fractions that involve infinity or division by zero, use the Number Crunch command instead of Simplify.

The operations performed depend on the type of expression selected, as shown in the following examples:

Performs integer arithmetic.

Selected expression

After simplifying

When the result is too large to calculate, it appears as Infinity.

Groups terms in a sum or product.

Selected expression

After simplifying

Divides out common factors in a fraction.

Selected expression

After simplifying

Use the Remove Negative Powers command to remove negative powers from the result.

Interprets the complex number i, represented by the Greek letter iota .

Selected expression

After simplifying

Distributes exponentiation across multiplication and division.

Selected expression

After simplifying

Simplifying  x produces  rather than .

Simplify Some

Simplifies the selected expression, as the Simplify command does, but does not multiply frac­tions.

Selected expression

After simplifying

Isolate Term

Isolates a selected expression on one side of the equal sign. This command does not combine terms or solve for the selected expression.

Selected expression

After isolating term

Expand First Term

Expands the first term of a selected summation or product.

Selected expression

After expanding

Expand All Terms

Expands all terms of a selected summation or product.

Selected expression

After expanding

September 30, 2016