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

gr_060a.png 

Expanded

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Multiplied out

gr_060c.png 

Multiplied out again

gr_060d.png 

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

Original selection

gr_061a.png 

Expanded

gr_061b.png 

Value substituted

gr_061c.png 

Result computed

gr_061d.png 

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 Commands pop-up menu

Matrix Transpose

Transposes the rows and columns of a selected matrix.

Selected matrix

After transposing

gr_062a.png 

gr_062b.png 

Matrix Algebra

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

Selected matrix (or matrices)

After matrix algebra

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gr_063b.png 

gr_063b.png 

gr_063d.png 

gr_063e.png 

gr_063f.png 

gr_063g.png 

gr_063h.png 

gr_063i.png 

gr_063j.png 

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

Addition pop-up menu

Add Fractions

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

Selected fraction

After adding

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gr_064b.png 

gr_064c.png 

gr_064d.png 

gr_064e.png 

gr_064f.png 

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.

Selected expression

gr_065a.png 

Added once

gr_065b.png 

Added twice

gr_065c.png 

Order Sum

Arranges polynomials in decreasing powers of a selected variable.

Before ordering

After ordering

gr_066a.png 

gr_066b.png 

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

gr_067a.png 

gr_067b.png 

Multiplication pop-up menu

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

gr_068a.png 

gr_068b.png 

gr_068c.png 

gr_068d.png 

Factor Some

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

Before factoring

After factoring

gr_069a.png 

gr_069b.png 

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

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gr_070b.png 

gr_070c.png 

gr_070d.png 

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

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gr_071b.png 

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:

Distributes division over addition.

Selected expression

After distributing

gr_072a.png 

gr_072b.png 

Use the Add Fractions command to perform the reverse operation.

Distributes multiplication over addition.

Selected expression

After distributing

gr_073a.png 

gr_073b.png 

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

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gr_074b.png 

gr_074c.png 

gr_074d.png 

gr_074e.png 

gr_074f.png 

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

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gr_075b.png 

gr_075c.png 

gr_075d.png 

gr_075e.png 

gr_075f.png 

Division pop-up menu

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

gr_076a.png 

Divided once

gr_076b.png 

Divided twice

gr_076c.png 

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

gr_077a.png 

gr_077b.png 

gr_077c.png 

gr_077d.png 

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

gr_078a.png 

gr_078b.png 

gr_078c.png 

gr_078d.png 

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

gr_079a.png 

gr_079b.png 

gr_079c.png 

gr_079d.png 

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

gr_080a.png 

gr_080b.png 

gr_080c.png 

gr_080d.png (no effect)

Evaluation pop-up menu

Number Crunch

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

Selected expression

After number crunch

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gr_081b.png 

gr_081c.png 

gr_081d.png 

gr_081e.png 

gr_081f.png 

gr_081g.png 

gr_081h.png 

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

gr_082a.png 

gr_082b.png 

gr_082c.png 

gr_082d.png 

Show All Digits

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

Selected number

After showing all digits

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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

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gr_084b.png 

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

Selected expression

After evaluating

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gr_085b.png 

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

Selected expression

After evaluating

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gr_086b.png 

Extracts a term from a sum or product.

Selected expression

After evaluating

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gr_087b.png 

Evaluates a substitution just as the Evaluate Substitution command does.

Rewrites an Evaluate Between operator as a difference.

Selected expression

After evaluating

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gr_088b.png 

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

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gr_089b.png 

gr_089c.png 

gr_089d.png 

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

Selected expression

After evaluating

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gr_090b.png 

Rewrites a choice function in terms of factorials.

Selected expression

After evaluating

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gr_091b.png 

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

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gr_092b.png 

gr_092c.png 

gr_092d.png 

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

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Evaluated once

gr_093b.png 

Evaluated twice

gr_093c.png 

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

gr_094a.png 

gr_094b.png 

gr_094c.png 

gr_094d.png 

Evaluate Derivatives 1 Level

Evaluates only the first level of a selected derivative.

Before

After

gr_095a.png 

gr_095b.png 

Rules pop-up menu

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

gr_096a.png 

Selected expression

gr_096b.png 

After rule is applied

gr_096c.png 

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

gr_097a.png 

Before rule is applied

gr_097b.png 

After rule is applied

gr_097c.png 

Other Rewrites pop-up menu

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

gr_098a.png 

gr_098b.png 

gr_098c.png 

gr_098d.png 

gr_098e.png 

gr_098f.png 

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

Groups terms in a sum or product.

Selected expression

After simplifying

gr_099a.png 

gr_099b.png 

gr_099c.png 

gr_099d.png 

gr_099e.png 

gr_099f.png 

Divides out common factors in a fraction.

Selected expression

After simplifying

gr_100a.png 

gr_100b.png 

gr_100c.png 

gr_100d.png 

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

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

Selected expression

After simplifying

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gr_101b.png 

Distributes exponentiation across multiplication and division.

Selected expression

After simplifying

gr_102a.png 

gr_102b.png 

gr_102c.png 

gr_102d.png 

Simplifying simp1.png x produces simp2.png rather than simp3.png.

Simplify Some

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

Selected expression

After simplifying

gr_103a.png 

gr_103b.png 

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

gr_104a.png 

gr_104b.png 

gr_104c.png 

gr_104d.png 

gr_104e.png 

gr_104f.png 

Expand First Term

Expands the first term of a selected summation or product.

Selected expression

After expanding

gr_105a.png 

gr_105b.png 

gr_105c.png 

gr_105d.png 

Expand All Terms

Expands all terms of a selected summation or product.

Selected expression

After expanding

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gr_106b.png 

gr_106c.png 

gr_106d.png 


September 30, 2016

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