Question

How do you simplify `\sqrt { 225x ^ { 3} y ^ { 9} }`?

Answer 1

`15xy^4sqrt(xy)`

Explanation:

We must find as many squares as possible inside the square root: for the numeric part, we have `225 = 15^2`, which is already a square. For the variable part, we have `x^3 = x^2\cdot x`, and `y^9=y^8\cdot y`. Note that every variable raised to an even power is a square, since `x^(2k) = (x^k)^2`.

Now, use the fact that the square root of a multiplication is the multiplications of the square roots: `sqrt(ab)=sqrt(a)sqrt(b)` to get

`sqrt(225x^3y^9) = sqrt(15^2\cdot x^2\cdot x \cdoty^8\cdot y) = sqrt(255)sqrt(x^2)sqrt(x)sqrt(y^8)sqrt(y)`

Simplify the squares with the square roots:

`sqrt(255)sqrt(x^2)sqrt(x)sqrt(y^8)sqrt(y) = 15x\sqrt(x)y^4sqrt(y)`

Rearrange for better visualization:

`15x\sqrt(x)y^4sqrt(y) = 15xy^4sqrt(xy)`