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2011-02-01T00:00:01
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<p>The absurdity of it!<br></br>
<br></br>
Find the exact values of $$\sqrt{2 + \sqrt 3}- \sqrt{2 - \sqrt3}$$ and of $$\root 3 \of {2 + \sqrt 5}+ \root 3 \of {2 - \sqrt 5}$$<br></br>
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Congratulations to Hyeyoun from St Paul's Girls School, London, to
Sue Liu of Madras College, St Andrew's, Scotland, Sanjay from The
Perse School, Cambridge and Bill from Alcester Grammar School for
your solutions.<br></br>
<br></br>
We take the square root symbol in the question to signify the
positive square root. The tactic here is to square both sides and
then find the correct square root. If
$\sqrt{2+\sqrt{3}}-\sqrt{2-\sqrt{3}}=X$, then $$X^2=
\left(\sqrt{2+\sqrt{3}}-\sqrt{2-\sqrt{3}}\right)
\left(\sqrt{2+\sqrt{3}}-\sqrt{2-\sqrt{3}}\right).$$ The right hand
side equals $$2 + \sqrt{3}-2\left(\sqrt{2+\sqrt{3}}\times
\sqrt{2-\sqrt{3}}\right) +2-\sqrt{3},$$ and
$$\sqrt{2+\sqrt{3}}\times\sqrt{2-\sqrt{3}}=1.$$ Therefore $$X^2 =
2+\sqrt{3}-2+2-\sqrt{3}=2.$$ Does $X=-\sqrt{2}$ or
$+\sqrt{2}$?<br></br>
<br></br>
Well $2+\sqrt{3}> 2-\sqrt{3}$, so $\sqrt{2+\sqrt{3}}>
\sqrt{2-\sqrt{3}}$, so $X$ is positive and we have
$X=\sqrt{2}$.<br></br>
<br></br>
Note that we could take each square root to be positive or
negative. If so, then the question is much harder and there are
more solutions for $X$. For example, we could take $\sqrt{3} =
1{\cdot}732\cdots$; then $\sqrt{2+\sqrt{3}}$ has two values
(approximately $\pm 1{\cdot}93$), and $\sqrt{2-\sqrt{3}}$ has two
values (approximately $\pm 0{\cdot}52$). It follows that $X$ has
four values (approximately $\pm 2{\cdot}45$ and $\pm 1{\cdot}41$).
Alternatively, we could take $\sqrt{3} = -1{\cdot}732\cdots$, and
then we would get even more solutions. <br></br>
<br></br>
In the second part there are again many solutions (because square
roots have two values and cube roots have three values). To
simplify the solution we restrict ourselves to real cube roots. We
want to find $$X = {\root 3\of {2+\sqrt{5}}} + {\root 3\of
{2-\sqrt{5}}}.$$ One way to do this is to write $a = {\root 3\of
{2+\sqrt{5}}}$ and $b= {\root 3\of {2-\sqrt{5}}}$, and use the
equation $$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 = a^3 + 3ab(a + b)
+ b^3.$$ As $X = a + b$, we have $$X^3 = (2 + \sqrt{5}) + 3X{\root
3\of {(2 + \sqrt{5})(2 - \sqrt{5})}} +(2 - \sqrt{5}).$$ As $\root
3\of {(2 + \sqrt{5})(2 - \sqrt {5}} = \root 3\of {-1} = -1$ this
gives $X^3 + 3X - 4 = 0$ and hence $$(X - 1)(X^2 + X + 4) = 0.$$ As
$X^2 + X + 4 = 0$ has only complex solutions, and we are looking
for the real values of $X$, so we have $X = 1$, that is $${\root
3\of {2 + \sqrt{5}}} + {\root 3\of {2 - \sqrt{5}}} = 1.$$ You can
check this on your calculator!<br></br>
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<mdoxml version="1.0"><span style="font-weight: bold;">Why do this problem?</span><br></br>
<br></br>
The problem provides good practice in the manipuation of surds and
in algebra (involving the expansions of $(p+q)^2$ and $(p+q)^3$,
the difference of two squares and the use of the Remainder Theorem
to factorise a cubic equation). If care is taken with the algebra
the result comes out in a satisfyingly neat way. The question looks
complicated but it turns out to be simple. <br></br>
<br></br>
<span style="font-weight: bold;">Possible approach</span>
<br></br>
<div>Although this is a longer, two part, question, the Hint gives
sufficient guidance for this to be set to a class to work on
independently.</div>
<br></br>
<div><span style="font-weight: bold;">Key questions</span></div>
<br></br>
<div>If 'extra' solutions are introduced by squaring or cubing, how
do you decide which are the correct solutions?</div>
<br></br>
<div style="font-weight: bold; font-style: italic;">This is how the
problem was used by Peter Thomas, a Sixth Form College
teacher:</div>
<br></br>
<div>I was absent at a meeting and set the class work to
consolidate topics taught the previous lesson. The work was routine
exercises from a textbook (Emanuel and Wood) which I encouraged
them to approach selectively (what I called 'bread and butter' with
some specific questions as a 'doggy bag' for homework).</div>
<br></br>
<div>Alongside this I set them the four nrich problems as 'cake'
with the instruction to tackle at least one.</div>
<br></br>
<div>Ab Surd Ity (this problem)</div>
<div>See also <a href="http://nrich.maths.org/public/viewer.php?obj_id=1838">Power
Quady</a></div>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=543">System
Speak</a></div>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=370">Root to
Poly</a></div>
<br></br>
<div>Then over the following couple of weeks the problems were
discussed in lessons as they related to the topics being
covered.</div>
<br></br>
<div>The reaction was positive. The teacher covering the lesson
reported strong engagement. From the individuals he mentioned this
extended to those who liked to finish quickly then sit back. When
the problems were discussed subsequently the contributions made
suggested most had had a go and got somewhere with at least one
problem.</div>
<br></br>
<div>It livened up what could have been a rather boring lesson of
consolidation.</div>
<br></br>
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For the first part try squaring the whole expression.<br></br>
<br></br>
For the second part, you could write $a = {\root 3\of
{2+\sqrt{5}}}$ and $b= {\root 3\of {2-\sqrt{5}}}$, and use the
identity $$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 = a^3 + 3ab(a + b)
+ b^3.$$ <br></br></mdoxml>
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Ab Surd Ity
Find the value of sqrt(2+sqrt3)-sqrt(2-sqrt3)and then of
cuberoot(2+sqrt5)+cuberoot(2-sqrt5).
Numbers and the Number System
Surds
Numbers and the Number System
Powers & roots
Algebra
Identities
Collections
epsilons
sfh10
Stage 5 - Reviewed 2012