Question

show that ∛6 is not a rational number.​

Answer 1

Answer:

let if possible 3√6 is rational . Then,

3√6 is rational,1/3 is rational

1/3*3√6 is rational( product of two rational is rational)

√6 is rational

This contradicts the fact that √6 is irrational.

This contradiction arrises by assuming that 3√6 is rational.

Thus,3√6 is irrational.

Step-by-step explanation:

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Answer 2

Let us assume ∛6 is a rational number.

Then, ∛6 = p/q ( where p and q are co - primes and q is not equal to zero )

★ Taking cube on both sides,we get:

$\sf{ \dfrac{p {}^{3} }{q {}^{3} } } = 6$....( 1 )

Now,

1³ = 1 and 2³ = 8

Thus,

1 < 6 < 8

⇒ 1 < p³/q³ < 8

⇒ 1 < p³/q³ < 2

From ( 1 )

p³/q³ = 6

⇒ 6q² = p³/q .....( 2 )

As q is an integer ⇒ 6q² is an integer ..( 3 )

Since, p and q have no common factor.

⇒ p³ and q will have no common factor.

⇒ p³/q is a fraction..........( 4 )

From ( 2 ),( 3 ) and ( 4 )

Integer = Fraction

Our assumption is wrong.

Hence,∛6 is not a rational number.