What should I do?

Nothing ;-).

Let's start from the binomial formula :

```
sage: a, b, c, i, j = var("a, b, c, i, j")
sage: BF=(a+b)^i==sum(a^j*b^(i-j)*binomial(i, j), j, 0, i, hold=True) ; BF
(a + b)^i == sum(a^j*b^(i - j)*binomial(i, j), j, 0, i)
```

($$ {\left(a + b\right)}^{i} = {\sum_{j=0}^{i} a^{j} b^{i - j} \binom{i}{j}} $$

Let's generalize it by using wildcards :

```
sage: w0, w1, w2 = (SR.wild(u) for u in range(3))
sage: BFG = BF.subs({a:w0, b:w1, i:w2}) ; BFG
($1 + $0)^$2 == sum($1^($2 - j)*$0^j*binomial($2, j), j, 0, $2)
```

$$ {\left(\$1 + \$0\right)}^{\$2} = {\sum_{j=0}^{\$2} \$1^{\$2 - j} \$0^{j} \binom{\$2}{j}} $$

and use it on a general form of your original expression :

```
sage: Ex=(a + b)^i + (a - b)^i ; Ex
(a + b)^i + (a - b)^i
```

$$ {\left(a + b\right)}^{i} + {\left(a - b\right)}^{i} $$

```
sage: ExT=Ex.subs(BFG) ; ExT
sum(a^j*(-b)^(i - j)*binomial(i, j), j, 0, i) + sum(a^j*b^(i - j)*binomial(i, j), j, 0, i)
```

$$ {\sum_{j=0}^{i} a^{j} \left(-b\right)^{i - j} \binom{i}{j}} + {\sum_{j=0}^{i} a^{j} b^{i - j} \binom{i}{j}} $$

which can be rewritten ("contracted") as a single sum :

```
sage: ExTR=ExT.maxima_methods().sumcontract() ; ExTR
sum(a^j*(-b)^(i - j)*binomial(i, j) + a^j*b^(i - j)*binomial(i, j), j, 0, i)
```

$$ {\sum_{j=0}^{i} a^{j} \left(-b\right)^{i - j} \binom{i}{j} + a^{j} b^{i - j} \binom{i}{j}} $$

whose summand can be factored as :

```
sage: ExTR.operands()[0].collect_common_factors()
a^j*((-b)^(i - j) + b^(i - j))*binomial(i, j)
```

$$ a^{j} {\left(\left(-b\right)^{i - j} + b^{i - j}\right)} \binom{i}{j} $$

Now, assuming that $a,b\in\mathbb{R},\,b>0$ :

if $i-j$ is *odd*, $\left(\left(-b\right)^{i - j} + b^{i - j}\right)=-\left(b\right)^{i - j} + b^{i - j}=0$

if $i-j$ is *even*, $\left(\left(-b\right)^{i - j} + b^{i - j}\right)=2\,b^{i - j}$.

All non-zero terms of the sum are therefore products whose $b$ factor is at an *even* power ; therefore, if $b$ is a square root, these terms can all be simplified as a real *without* (external) radical.

That's (more or less) what Sage does when putting its computation results in its "preffered format", which has already been the cause of very many questions...

HTH,

Homework ?