## STACK Documentation

Documentation home | Category index | Site map# 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.

- Algebraic equivalence with the correct answer: use
`ATAlgEquiv`

. - 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:

- Matrix examples in showing working.
- An example of a question with
`simp:false`

is discussed in authoring quick start 7. - Generating random algebraic expressions which need to be "gathered and sorted".

Note also that question tests do not simplify test inputs.

Documentation home | Category index | Site map