Question

if you solve this problem i will definitely mark you as a brainlist ​

Answer 1

Question:-

• Prove that,$\sf \small \sqrt{ log_{3}(81)} + 5 \sqrt{ log_{7}(343) }$ is irrational.

Proof:-

$\sf \small \sqrt{ log_{3}(81)} + 5 \sqrt{ log_{7}(343) }$

$\sf \small = \sqrt{ log_{3}( {3}^{4} )} + 5 \sqrt{ log_{7}( {7}^{3} ) }$

$\sf \small = \sqrt{4} + 5 \sqrt{3 }$

$\sf \small =2 + 5 \sqrt{3 }$

Let us assume that,

$\sf \small 2 + 5 \sqrt{3 }$ is rational number, say r.

So,

$\sf \small 2 + 5 \sqrt{3 } = r$

$\sf \implies \small r - 2 = 5 \sqrt{3 }$

$\sf \implies \small \sqrt{3 } = \frac{r - 2}{ 5}$

As r is rational, r-2 is rational which means (r-2)/5 is rational.>> √3 is rational.

But this contradicts the fact that √3 is rational.

Hence, our assumption is wrong.

So,

$\sf \small 2 + 5 \sqrt{3 }$ is irrational.

Or,

$\sf \small \sqrt{ log_{3}(81)} + 5 \sqrt{ log_{7}(343) }$ is irrational.

Answer 2

Answer:

Question:-

Prove that,\sf \small \sqrt{ log_{3}(81)} + 5 \sqrt{ log_{7}(343) }

log

3

(81)

+5

log

7

(343)

is irrational.

Proof:-

\sf \small \sqrt{ log_{3}(81)} + 5 \sqrt{ log_{7}(343) }

log

3

(81)

+5

log

7

(343)

\sf \small = \sqrt{ log_{3}( {3}^{4} )} + 5 \sqrt{ log_{7}( {7}^{3} ) }=

log

3

(3

4

)

+5

log

7

(7

3

)

\sf \small = \sqrt{4} + 5 \sqrt{3 }=

4

+5

3

\sf \small =2 + 5 \sqrt{3 }=2+5

3

Let us assume that,

\sf \small 2 + 5 \sqrt{3 }2+5

3

is rational number, say r.

So,

\sf \small 2 + 5 \sqrt{3 } = r2+5

3

=r

\sf \implies \small r - 2 = 5 \sqrt{3 }⟹r−2=5

3

\sf \implies \small \sqrt{3 } = \frac{r - 2}{ 5}⟹

3

=

5

r−2

As r is rational, r-2 is rational which means (r-2)/5 is rational.>> √3 is rational.

But this contradicts the fact that √3 is rational.

Hence, our assumption is wrong.

So,

\sf \small 2 + 5 \sqrt{3 }2+5

3

is irrational.

Or,

\sf \small \sqrt{ log_{3}(81)} + 5 \sqrt{ log_{7}(343) }

log

3

(81)

+5

log

7

(343)

is irrational.

hope it helps