What is the square root of 135bc^2d^3 times the square root of 5b^2d?

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`sqrt(135bc^2d^3)*sqrt(5b^2d)` simplifies to `15bcd^2sqrt(3b)` .

Assuming all of the variables are nonnegative, we use a property of square roots; namely `sqrt(ab)=sqrt(a)sqrt(b)`, or the square root of a product can be written as the product of square roots. Thus:

`sqrt(135bc^2d^3)*sqrt(5b^2d)=sqrt(675b^3c^2d^4)`

To simplify a radical expression (in this case, a square root) there cannot be any fractions in the radicand (inside the radical sign) or a radical in the denominator. Neither of these happens here. The other restriction is that there can be no perfect square factors in the radicand (generally no perfect nth power factors in an nth root radical, but here, n=2, so we are concerned with square factors).

`675=3^3*5^2=3(3^2*5^2)=3*225=3*(15)^2`
`b^3=b*(b)^2`
`c^2=(c)^2`
`d^4=(d^2)^2`

We can rewrite as: `sqrt(675b^3c^2d^4)=sqrt(225b^2c^2d^4)*sqrt(3b)`

Then, `15bcd^2sqrt(3b)`

Note that without the condition that all of the variables were nonnegative, we cannot apply the property that `sqrt(a)sqrt(b)=sqrt(ab)` . Another thing to note is that a radical written without an index is taken to be the square root; the way to indicate another root is to include the index.

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