Question

Prove that each of the following
is irration3√7​

Answer 1

$\large{\boxed{\bf{To\ Prove\ :- }}}$

• 3√7 is an irrational number.

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Proof :-

Let us assume that 3√7 is rational. i.e it's not an irrational number.

• Let 3√7 = a/b ,

Where a and b are integers.

Now

• 3√7 = a/b

=> √7 = a/b × 1/3

=> √7 = a/3b

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➪ In RHS a, b and 3 are rational numbers.

➪ In LHS √7 is also rational.

But it contradicts the fact that√7 is irrational.

So, our assumption is wrong.

Hence, 3√7 is an irrational number.

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Answer 2

• 3√7 ɪs ᴀɴ ɪʀʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ.

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ᴘʀᴏᴏғ :-

ʟᴇᴛ ᴜs ᴀssᴜᴍᴇ ᴛʜᴀᴛ 3√7 ɪs ʀᴀᴛɪᴏɴᴀʟ. ɪ.ᴇ ɪᴛ's ɴᴏᴛ ᴀɴ ɪʀʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ.

• ʟᴇᴛ 3√7 = ᴀ/ʙ ,

ᴡʜᴇʀᴇ ᴀ ᴀɴᴅ ʙ ᴀʀᴇ ɪɴᴛᴇɢᴇʀs.

ɴᴏᴡ

• 3√7 = ᴀ/ʙ

=> √7 = ᴀ/ʙ × 1/3

=> √7 = ᴀ/3ʙ

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➪ ɪɴ ʀʜs ᴀ, ʙ ᴀɴᴅ 3 ᴀʀᴇ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs.

➪ ɪɴ ʟʜs √7 ɪs ᴀʟsᴏ ʀᴀᴛɪᴏɴᴀʟ.

ʙᴜᴛ ɪᴛ ᴄᴏɴᴛʀᴀᴅɪᴄᴛs ᴛʜᴇ ғᴀᴄᴛ ᴛʜᴀᴛ√7 ɪs ɪʀʀᴀᴛɪᴏɴᴀʟ.

sᴏ, ᴏᴜʀ ᴀssᴜᴍᴘᴛɪᴏɴ ɪs ᴡʀᴏɴɢ.

ʜᴇɴᴄᴇ, 3√7 ɪs ᴀɴ ɪʀʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ.

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