Question

Simplify


`\sqrt{81-18√(7)+7 }`

Answer 1

The given equation is expansion of (a-b)² = a²-2ab+b²

`\sqrt{81-18√(7)+7} \\ \sqrt{ {9}^(2) -2 * 9 * √(7)+ { √(7) }^(2) } \\ \sqrt{ {(9 - √(7))}^(2) } \\ {(9 - √(7))}`

hope that help.

Answer 2

`9-√(7)`

Explanation:

Given expression:
`\sqrt{81-18√(7)+7}`

To simplify the given expression, use the "A minus B Whole Square" identity.

A minus B Whole Square identity

`(a-b)^2=a^2-2ab+b^2`

Comparing
`81-18√(7)+7` to the right side of the identity:

• 
  `a^2=81 \implies a=√(81)=9`
• 
  `b^2=7\implies b=√(7)`
• 
  `-2ab=-18√(7)=-2 \cdot 9√(7)`

Therefore:

`(9-√(7))^2=81-18√(7)+7`

Finally:

`\sqrt{81-18√(7)+7}=\sqrt{(9-√(7))^2}=9-√(7)`