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Simplification & ordering

Algebraic equivalence

Is \((a^x)^y \equiv a^{x\,y}\)? Well, it depends! In particular you can easily derive the contradiction \[ -1 = (-1)^1 = (-1)^{2\times \frac{1}{2}} \] and using our rule \((a^x)^y \equiv a^{x\,y}\) \[ = \left({(-1)^{2}}\right)^{\frac{1}{2}} = 1^{\frac{1}{2}} = 1.\] To avoid problems like this we therefore have decided that

ATAlgEquiv((a^x)^y, a^(x*y)) = [0, ""]

If you are teaching rules of indices to students for the first time this might come as a surprise! If you would like STACK to implement this rule, then you need to also assume(a>0). This can be done in the feedback variables. This is a design decision and not a bug (and is recorded in the system unit tests)!

Ordering terms

Maxima chooses an order in which to write terms in an expression.
By default, this will use reverse lexicographical order for simple sums, so that we have \(b+a\) instead of \(a+b\). In elementary mathematics this looks a little odd! One way to overcome this is to use simplification below but another way is to alter the order in which expressions are transformed.

To alter the order in STACK you can use the Maxima commands orderless and ordergreat. To have \(a+b\) you can use

ordergreat(a,b);

See Maxima's documentation for more details.

Only one orderless or ordergreat command can be issued in any session. The last one encountered will be used and the others ignored.
No warnings or errors are issued if more than one is encountered.

Logarithms to an arbitrary base

By default, Maxima does not provide logarithms to an arbitrary base. To overcome this, STACK provides a function lg for student entry.

  • lg(x) is log of \(x\) to the base 10.
  • lg(x, a) is log of \(x\) to the base \(a\).

STACK provides no simplification rules for these logarithms. To simplify you must transform back to natural logarithms.

For example (with simp:true or simp:false)

p:lg(27, 3)
q:ev(p, lg=logbasesimp)

results in p=lg(27, 3), and q=3.

The algebraic equivalence function algebraic_equivalence, and so anything upon which it depends, will automatically remove logarithms to other bases.
This includes the answer tests as needed.

Selective simplification

The level of simplification performed by Maxima can be controlled by changing Maxima's global variable simp, e.g.

simp:true

This variable can be set at the question level using the options or for each Potential response tree.

When this is false, no simplification is performed and Maxima is quite happy to deal with an expression such as \(1+4\) without actually performing the addition. This is most useful for dealing with very elementary expressions.

If you are using simp:false to evaluate an expression with simplification on, you can use

ev(ex,simp)

Unary minus and simplification

There are still some problems with the unary minus, e.g. sometimes we get the display \(4+(-3x)\) when we would actually like to always display as \(4-3x\).
This is a problem with the unary minus function -(x) as compared to binary infix subtraction a-b.

To reproduce this problem type in the following into a Maxima session:

simp:false;
p:y^3-2*y^2-8*y;

This displays the polynomial as follows.

y^3-2*y^2+(-8)*y

Notice the first subtraction is fine, but the second one is not. To understand this, we can view the internal tree structure of the expression by typing in

?print(p);
((MPLUS) ((MEXPT) $Y 3) ((MMINUS) ((MTIMES) 2 ((MEXPT) $Y 2))) ((MTIMES) ((MMINUS) 8) $Y))

In the structure of this expression the first negative coefficient is -(2*y^2) BUT the second is -(8)*y.
This again is a crucial but subtle difference!
To address this issue we have a function

unary_minus_sort(p);

which pulls "-" out the front in a specific situation: that of a product with a negative number at the front.
The result here is the anticipated y^3-2*y^2-8*y.

Note that STACK's display functions automatically apply unary_minus_sort(...) to any expression being displayed.

If you really insist on a kludge....

In some situations you may find you really do need to work at the display level, construct a string and display this to the student in Maxima. Please avoid doing this!

a:sin(x^2);
b:1+x^2;
f:sconcat("\\frac{",stack_disp(a,""),"}{",stack_disp(b,""),"}");

Then you can put in \({@f@}\) into one of the CASText fields. Note, you need to add LaTeX maths delimiters, because when the CAS returns a string the command {@f@} will just display the contents of the string without maths delimiters.

Tips for manipulating expressions

How do we do the following in Maxima? \[ (1-x)^a \times (x-1) \rightarrow -(1-x)^{a+1}.\] Try

q:(1-x)^a*(x-1);
q:ratsubst(z,1-x,q);
q:subst(z=1-x ,q);

How do we do the following in Maxima? \[ (x-1)(k(x-1))^a \rightarrow (x-1)^{a+1}k^a.\]

 factor(radcan((x-1)*(k*(x-1))^a))

Maxima's internal representation of an expression sometimes does not correspond with what you expect -- in that case, dispform may help to bring it into the form you expect. For example, the output of solve in the following code shows the \(b\) in the denominator as \(b^{-1}\) which gives unnatural-looking output when a value is substituted in -- this is fixed by using dispform and substituting into that variants instead.

simp:true;
eqn:b = 1/(6*a+3);
ta1: expand(rhs(solve(eqn,a)[1]));
dispta1:dispform(ta1);
simp:false;
subst(2,b,ta1);
subst(2,b,dispta1);

Creating sequences and series

One problem is that makelist needs simplification. To create sequences/series, try something like the following

an:(-1)^n*2^n/n!
N:8
S1:ev(makelist(k,k,1,N),simp)
S2:maplist(lambda([ex],ev(an,n=ex)),S1)
S3:ev(S2,simp)
S4apply("+",S3)

Of course, to print out one line in the worked solution you can also apply("+",S2) as well.

To create the binomial coefficients

simp:false;
n:5;
apply("+",map(lambda([ex],binomial(n,ex)*x^ex), ev(makelist(k,k,0,5),simp)));

Surds

Imagine you would like the student to expand out \( (\sqrt{5}-2)(\sqrt{5}+4)=2\sqrt{5}-3 \). There are two tests you probably want to apply to the student's answer.

  1. Algebraic equivalence with the correct answer: use ATAlgEquiv.
  2. That the expression is "expanded": use ATExpanded.

You probably then want to make sure a student has "gathered" like terms. In particular you'd like to make sure a student has either \[ 2\sqrt{5}-3 \mbox{ or } \sqrt{20}-3\] but not \[ 5+4\sqrt{2}-2\sqrt{2}+6.\] This causes a problem because ATComAss thinks that \[ 2\sqrt{5}-3 \neq \sqrt{20}-3.\] So you can't use ATComAss here, and guarantee that all random variants will work by testing that we really have \(5+4\sqrt{2}\) for example.

What we really want is for the functions sqrt and + to appear precisely once in the student's answer, or that the answer is a sum of two things.

Control of surds

See also the Maxima documentation on radexpand. For example

radexpand:false$
sqrt((2*x+10)/10);
radexpand:true$
sqrt((2*x+10)/10);

The first of these does not pull out a numerical denominator. The second does.

Boolean functions

See the page on propositional logic.

Further examples

Some further examples are given elsewhere:

Note also that question tests do not simplify test inputs.

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