Question

Check whether 3√18 is a rational or irrational number.

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Step by step explaination required!!

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Answer 1

$\Large{\underline{\underline{\sf{Required\:Answer}}}}$

$\large{\boxed{\sf{\red{ 3 \sqrt{18}\:is\:an\:irrational\:number}}}}$

$\Large{\underline{\underline{\sf{Explanation:}}}}$

So, here we needed to identify whether the given number is a rational number or an irrational number.

To do so, we first of all,

• Factorise the number which is under the root.

$\begin{array}{r | l} 2 & 18 \\ 3 & 9 \\ 3 & 3 \\ & 1 \end{array}$

$\displaystyle{\sf{18 = 2\times 3 \times 3}}$

• Simplify the square roots by observing the perfect square.

$\implies{\sf{ 3\sqrt{18} = 3 ( \sqrt{2} \times \sqrt{3} \times \sqrt{3})}}$

$\implies{\sf{ = 3 \times 3 \times \sqrt{2}}}$

$\implies{\sf{ = 9 \sqrt{2}}}$

• If root is not completely dissolved then it is irrational else it is rational.

$\displaystyle{\sf{ \sqrt{2}\:is\:an\:irrational\:number}}$

$\displaystyle{\underline{\sf{\red{\therefore\:3 \sqrt{18}\:is\:an\:rational\:number.}}}}$

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Answer 2

Solution :-

❥ Lєt us αssumє , tσ thє cσntrαrч , thαt 3√18 ís rαtíσnαl.

thαt ís , wє cαn fínd cσprímє α αnd в ( в ≠ 0 ) such thαt 3 √18 = α/b.

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Thєrєfσrє ,

We know the factors of 18 = 2 × 3 × 3

Hence , from this we get :-

3√ 18 = 3 ( √2 × √3 × √3 )

= 3 × 3 × √2

= 9√2

Now,

Let us assume that,

√2 is a rational number of simplest form $\frac{a}{b}$, having no common factor other than 1.

 

√2 = $\frac{a}{b}$

On squaring both sides, we get ;

        2 = $\frac{a^{2}}{b^{2}}$

   ❥ a² = 2b²

Clearly, a² is divisible by 2.

So, a is also divisible by 2.

Now, let some integer be c.

❥ a = 2c

Substituting for a, we get ;

❥ 2b² = 2c

Squaring both sides,

❥ 2b² = 4c²

❥ b² = 2c²

❧ Thís mєαns thαt, 2 dívídєs в², αnd sσ 2 dívídєs в.

thєrєfσrє, α αnd в hαvє αt lєαst 2 αs α cσmmσn fαctσr. thís cσntrαdícts .

So, √2 is irrational.

thís cσntrαdíctíσn hαs αrísєn вєcαusє of σur íncσrrєct

αssumptíσn thαt 3√18 ís rαtíσnαl.

Sσ , Wє cσncludє thαt 3√18 ís írrαtíσnαl.

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Extra shots :-

❃ What is Rational Number ?

❥ A rαtíσnαl numвєr ís α numвєr thαt cαn вє єхprєss αs thє rαtíσ σf twσ íntєgєrs. α numвєr thαt cαnnσt вє єхprєssєd thαt wαч ís írrαtíσnαl.

❃ What is Irrational Number ?

❥ Thє írrαtíσnαl numвєrs αrє αll thє rєαl numвєrs whích αrє nσt rαtíσnαl numвєrs. thαt ís, írrαtíσnαl numвєrs cαnnσt вє єхprєssєd αs thє rαtíσ σf twσ íntєgєrs. ... ín thє cαsє σf írrαtíσnαl numвєrs, thє dєcímαl єхpαnsíσn dσєs nσt tєrmínαtє, nσr єnd wíth α rєpєαtíng sєquєncє.

ྉ How to prove a number be Irrational ?

❥ Thє usuαl αpprσαch ís “prσσf вч cσntrαdíctíσn” – σnє σf thє mσst pσwєrful αnd usєful prσσf tєchníquєs ín mαthєmαtícs. чσu stαrt вч αssumíng thαt α numвєr ís rαtíσnαl, αnd thєn shσw thαt thís lєαds tσ α lσgícαl cσntrαdíctíσn. thís dєmσnstrαtєs thαt чσur ínítíαl αssumptíσn must вє fαlsє, sσ thє numвєr must вє írrαtíσnαl.

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