Question

How do you write the expression `sqrt(48x^6y^3)` in simplified radical form.

Answer 1

First of all, let's deal with the numeric part: by factoring 48 with prime numbers, `\sqrt{48}=\sqrt{2^4 \cdot 3}`. Now use the fact that the square root of a product is the product of the square roots, so we have `\sqrt{48}=\sqrt{16}\cdot \sqrt{3}=4\sqrt{3}`

As for the variables, we can again say that `\sqrt{x^6y^3}=\sqrt{x^6}\sqrt{y^3}`. You can write `x^6` as `x^2\cdot x^2\cdot x^2`, so `\sqrt{x^6}=\sqrt{x^2\cdot x^2\cdot x^2}=\sqrt{x^2}\cdot\sqrt{x^2}\cdot\sqrt{x^2}=x\cdot x \cdot x =x^3`

The same reason bring us to write `y^3` as `y^2 \cdot y`, so `\sqrt{y^3}=\sqrt{y^2\cdot y}=\sqrt{y^2}\cdot\sqrt{y}=y\sqrt{y}`.

Putting all the pieces together, we have that
`\sqrt{48x^6y^3}=4x^3y\sqrt{3y}`