Question

Which expression is equivalent to square root 55x^7y6/11x^11y^8

Answer 1

`\large\boxed{\sqrt{(55x^7y^6)/(11x^(11)y^8)}=(\sqrt5)/(x^2y)}`

Explanation:

`\sqrt{(55x^7y^6)/(11x^(11)y^8)}=\sqrt{(55)/(11)\cdot(x^7)/(x^(11))\cdot(y^6)/(y^8)}\qquad\text{use}\ (a^n)/(a^m)=a^(n-m)\\\\=\sqrt{5x^(7-11)y^(6-8)}=\sqrt{5x^(-4)y^(-2)}\qquad\text{use}\ √(ab)=√(a)\cdot√(b)\\\\=\sqrt5\cdot\sqrt{x^(-4)}\cdot\sqrt{y^(-2)}=\sqrt5\cdot\sqrt{x^((-2)(2))}\cdot\sqrt{y^((-1)(2))}\qquad\text{use}\ (a^n)^m=a^(nm)\\\\=\sqrt5\cdot\sqrt{(x^(-2))^2}\cdot\sqrt{(y^(-1))^2}\qquad\text{use}\ √(a^2)=a`

`=\sqrt5\cdot x^(-2)\cdot y^(-1)\qquad\text{use}\ a^(-n)=(1)/(a^n)\to a^(-1)=(1)/(a)\\\\=\sqrt5\cdot(1)/(x^2)\cdot(1)/(y)=(\sqrt5)/(x^2y)`