Question

Simplify this radical.
Square root of x to the 13th power

Answer 1

`\large\boxed{\sqrt{x^(13)}=x^6√(x)}`

Explanation:

`\sqrt{x^(13)}=\sqrt{x^(12+1)}\qquad\text{use}\ a^na^m=a^(n+m)\\\\=\sqrt{x^(12)x^1}\qquad\text{use}\ √(ab)=√(a)\cdot√(b)\\\\=\sqrt{x^(12)}\cdot√(x)=\sqrt{x^(6\cdot2)}\cdot√(x)\qquad\text{use}\ (a^n)^m=a^(nm)\\\\=√((x^6)^2)\cdot√(x)\qquad\text{use}\ √(x^2)=|x|\\\\=|x^6|√(x)=x^6√(x)\ \text{because}\ x^6\geq0`

Answer 2

x^6
`√(x)`

Explanation:

we have:

`\sqrt{x^(13)}`

we can write:

`\sqrt{x^(13)}` :
`\sqrt{x^(2)*x^(2)*x^(2)*x^(2)*x^(2)*x^(2)*x}`

we know:

`\sqrt{x^(2)} = x`

so we have:

`\sqrt{x^(13)}` =
`\sqrt{x^(2)*x^(2)*x^(2)*x^(2)*x^(2)*x^(2)*x}`

we have:

`\sqrt{x^(2)*x^(2)*x^(2)*x^(2)*x^(2)*x^(2)*x}`=

`x^(6) √(x)`