Simplify  `sqrt(135b^2c^3d)`  *`sqrt(5b^2d)`                              

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violy | High School Teacher | (Level 1) Associate Educator

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What we do is multiply the radicands and write it in one radical sign first.

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

We can rewrite last radical expression as:

`sqrt(675)*sqrt(b^4)*sqrt(c^3)*sqrt(d^2)`

For the 675, we can factor it which one of the factors is a perfect square number. 

Perfect square numbers are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256,..

We will identify the largest perfect square number that can go into 675.

We can see that the last digit of 675 is 5. So, it is divisible by 5, or multiples of 5 like 15, 25, etc.

Hence, let us try 25 and 225.

675/25= 27  and 675/225 =3

The 27 can be still factor with one number being a perfect square. While, 3 can't be factor any further. Hence, the largest perfect square number that can go into 675 is 225. 

So, we will have:

`sqrt(675) = sqrt(225)*sqrt(3) = 15sqrt(3)`

For the variables, we divide the exponents of it by 2 (by 2 since its square root, which means we raised it by 1/2). The remainder will be the exponent of the remaining variable inside the radical sign.

`sqrt(b^4) = b^(4/2) = b^2`

`sqrt(c^3) = c^(3/2) = csqrt(c)`

`sqrt(d^2) = d^(2/2) = d`

So, we will have:

`15sqrt(3)*b^2*csqrt(c)*d = 15b^2cdsqrt(3c)`

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