How do you simplify $\sqrt { 225x ^ { 18} y ^ { 22} }$ ?

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Answer 1 · 2016-12-11T04:35:46

$15x^9y^11$

Recall : $sqrtx=x^(1/2), and sqrt(x^2)=(x^2)^(1/2)=x^(2*1/2)=x$ ,

$sqrt(225x^18y^22)=sqrt(15*15*x^18*y^22)=(15^2*x^18*y^22)^(1/2)$
$=15^(2*1/2)*x^(18*1/2)*y^(22*1/2)$
$=15*x^9*y^11$

Answer 2 · 2016-12-11T09:07:09

$sqrt(225x^18y^22) = abs(15x^9y^11)$

Note that:

$225x^18y^22 = 15^2*(x^9)^2*(y^11)^2 = (15x^9y^11)^2$

So $15x^9y^11$ is a square root of $225x^18y^22$ .

We also find:

$(-15x^9y^11)^2 = 225x^18y^22$

So $-15x^9y^11$ is also a square root of $225x^18y^22$ .

Note that if $a >= 0$ then $sqrt(a)$ denotes the non-negative square root.

So if, for example, $x < 0$ and $y > 0$ then $15x^9y^11 < 0$ would be the wrong square root.

We can express that we want the non-negative square root by taking the modulus:

$sqrt(225x^18y^22) = abs(15x^9y^11)$

How do you simplify \sqrt { 225x ^ { 18} y ^ { 22} }\sqrt { 225x ^ { 18} y ^ { 22} }?