Prove that each of the following
is irration3√7
Answers
Answered by
39
• 3√7 is an irrational number.
____________________
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
____________________
➪ 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.
____________________
Answered by
3
• 3√7 ɪs ᴀɴ ɪʀʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ.
____________________
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
____________________
➪ 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.
____________________
Similar questions