Linked Questions

20
votes
6answers
18k views

Why eliminate radicals in the denominator? [rationalizing the denominator] [duplicate]

Why do all school algebra texts define simplest form for expressions with radicals to not allow a radical in the denominator. For the classic example, $1/\sqrt{3}$ needs to be "simplified" to $\sqrt{3}...
8
votes
3answers
443 views

Why do we need rationalisation? [duplicate]

While solving a problem I came to an answer $\frac{1}{\sqrt{3}+1}$. But this was not the answer. The answer was $\frac{\sqrt{3}-1}{2}$ which comes on rationalisation of my answer. Then I divided ...
3
votes
1answer
5k views

When to rationalize numerator and/or denominator? [duplicate]

Sometimes, we have to rationalize either the numerator or the denominator, and sometimes we can still work the problem without rationalizing. So, in some cases, rationalizing can be done, although it ...
4
votes
4answers
203 views

Rationalizing denominator - why? [duplicate]

In many Algebra textbooks, why rationalizing the denominator is defined to be the simplest form ? I try to understand it but I cannot.Why is $\frac{\sqrt 3}{3}$ simpler than $\frac1{\sqrt 3}$?
3
votes
2answers
123 views

Why do we need to rationalize fractions? [duplicate]

Teachers often take off points from students who write 1/sqrt(2) instead of sqrt(2)/2. Why do we need to write it as sqrt(2) / 2 ? Where did that convention come from? Do we need to even do it? Why do ...
1
vote
1answer
134 views

Why are surds put on the numerator in the final answer when it is a fraction. [duplicate]

I have learnt that a fraction with a surd in its most simplest form should have the surd in the numerator and not the denominator? Why is it convention not to leave the surd on the denominator? Is it ...
61
votes
7answers
8k views

Polynomial division: an obvious trick? [reducing mod $\textit{simpler}$ multiples]

The following question was asked on a high school test, where the students were given a few minutes per question, at most: Given that, $$P(x)=x^{104}+x^{93}+x^{82}+x^{71}+1$$ and, $$Q(x)=x^4+...
8
votes
4answers
323 views

Help solving a limit

While helping a friend out for an exam (last year of high school), I found an exercise that neither of us could solve. I've tried a couple of different approaches but nothing seemed to work. Could ...
9
votes
3answers
267 views

What is $\frac{1}{1+\sqrt[3]{2}}$ in $\mathbb{Q}(\sqrt[3]{2})$?

Since $\mathbb{Q}(\sqrt[3]{2})$ is a field, any number $\neq 0$ has a reciprocal. How then to write $\frac{1}{1+\sqrt[3]{2}}$ as a number $a + b\sqrt[3]{2} + c\sqrt[3]{4}$ with fractions $a,b,c \in ...
3
votes
2answers
1k views

In $\Bbb Z[\sqrt{2}]=\{a+b\sqrt{2}\rvert a,b∈\Bbb Z\}$, show that every element of the form $(3+2\sqrt{2})^n$ is a unit, where n is a positive integer

In $\Bbb Z[\sqrt{2}]=\{a+b\sqrt{2}\rvert a,b∈\Bbb Z\}$, show that every element of the form $(3+2\sqrt{2})^n$ is a unit, where n is a positive integer. My understanding of a unit is that if a is a ...
2
votes
2answers
846 views

Division of Complex Numbers

Ahlfors says that once the existence of the quotient $\frac{a}{b}$ has been proven, its value can be found by calculating $\frac{a}{b} \cdot \frac{\bar b}{\bar b}$. Why doesn't this manipulation show ...
1
vote
4answers
79 views

Rationalize nested radical expression $\frac{8}{\sqrt{2-\sqrt{\frac{5+\sqrt{5}}{2}}}}$

I have a college task to rationalize this fraction. $$\frac{8}{\sqrt{2-\sqrt{\frac{5+\sqrt{5}}{2}}}}$$ I do not know how to simplify this fraction. Please, explain how to remove the radical from ...
6
votes
1answer
91 views

Rationalizing denominator containing a root of a polynomial - why is this possible / why does it work? [duplicate]

The problem is to express the number $$1\over x^5 + 2x^4 + 3x^3 + 3x^2 + 2$$ using only rational numbers in the denominator, knowing that $x^5 + 2 = -2(x + 1)(x^3 + x^2 + x)$. (This is an example ...
2
votes
1answer
61 views

Do canonical forms serve only one purpose?

In this answer I wrote that perhaps the principal utility of canonical forms is to tell whether two things are equal. Is $\dfrac 1 {1-\sqrt 2}$ equal to $1+\sqrt2$, or $\dfrac 1 {2\sqrt 3}$ to $\...