Question

Prove that
✓5 is irrational
&
3+2✓5 is irrational​

Answer 1

Answer:

Heyy....

Given:

We need to prove that √5 is irrational

Proof:

Let us assume that √5 is a rational number.

Sp it t can be expressed in the form p/q where p,q are co-prime integers and q≠0

⇒√5=p/q

On squaring both the sides we get,

⇒5=p²/q²

⇒5q²=p² —————–(i)

p²/5= q²

So 5 divides p

p is a multiple of 5

⇒p=5m

⇒p²=25m² ————-(ii)

From equations (i) and (ii), we get,

5q²=25m²

⇒q²=5m²

⇒q² is a multiple of 5

⇒q is a multiple of 5

Hence, p,q have a common factor 5. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number

√5 is an irrational number

Given:3 + 2√5

To prove:3 + 2√5 is an irrational number.Proof:Letus assume that 3 + 2√5 is a rational number.Soit can be written in the form a/b3 + 2√5 = a/bHere a and b are coprime numbers and b ≠ 0Solving3 + 2√5 = a/b we get,=>2√5 = a/b – 3=>2√5 = (a-3b)/b=>√5 = (a-3b)/2bThis shows (a-3b)/2b is a rational number. But we know that But √5 is an irrational number.so it contradictsour assumption.Our assumption of3 + 2√5 is a rational number is incorrect.3 + 2√5 is an irrational numberHence proved.

I hope it helps u..

Answer 2

♧︎︎︎ ǫᴜᴇsᴛɪᴏɴ ♧︎︎

• Prove that 3 + 2√5 is an irrational number

♧︎︎︎ ᴀɴsᴡᴇʀ ♧︎︎︎

➪ᴡᴇ ʜᴀᴠᴇ ᴛᴏ ᴘʀᴏᴠᴇ 3+2√5 ɪs ɪʀʀᴀᴛɪᴏɴᴀʟ

ʟᴇᴛ ᴜs ᴀssᴜᴍᴇ ᴛʜᴇ ᴏᴘᴘᴏsɪᴛᴇ

ᴛʜᴀᴛ ɪs 3+2√5 ɪs ʀᴀᴛɪᴏɴᴀʟ

ʜᴇɴᴄᴇ,

• 3+2√5=a/b

• 2√5 =a/b -3

• 2√5 =a-3b/b

• √5=1/2 × a-3b/b

• here a-3b /2b is rational number

• but √5 is irrational

sɪɴᴄᴇ,ʀᴀᴛɪᴏɴᴀʟ ≠ɪʀʀᴀᴛɪᴏɴᴀʟ

ᴛʜɪs ɪs ᴀ ᴄᴏɴᴛʀᴅɪᴄᴛɪᴏɴ

ᴛʜᴇʀᴇ ғᴏʀᴇ ,ᴏᴜʀ ᴀssᴜᴍᴘᴛɪᴏɴ ɪs ɪɴᴄᴏʀʀᴇᴄᴛ

ʜᴇɴᴄᴇ 3+23+2√5 ɪs ɪʀʀᴀᴛɪᴏɴᴀʟ

ʜᴇɴᴄᴇ ᴘʀᴏᴠᴇᴅ

ɪ ʜᴏᴘᴇ ᴛʜɪs ʜᴇʟᴘs ᴜʜʜ

ᴍᴀʀᴋ.ᴀs ʙʀᴀɪɴʟᴇᴀsᴛ