Question #fde5d

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Question #fde5d

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Answer 1 · 2016-10-16T22:44:42

$\frac{\sqrt{21 x^{2} y}}{\sqrt{75 x y^{5}}} = \frac{\sqrt{7 x}}{5 y^{2}}$ $sqrt(21x^2y)/sqrt(75xy^5)=sqrt(7x)/(5y^2)$

Explanation:

We will use the property that $\sqrt{a} \sqrt{b} = \sqrt{a b}$ $sqrt(a)sqrt(b) = sqrt(ab)$ to eliminate the radical from the denominator. To do so, we will multiply the numerator and denominator by such a value as to make the denominator the square root of a perfect square. With that, we will apply $\sqrt{a^{2}} = a$ $sqrt(a^2) = a$ for $a > 0$ $a>0$ .

First, let's figure out what that value will be. Note that ${(x^{a})}^{b} = x^{a b}$ $(x^a)^b = x^(ab)$ .

$75 x y^{5} = 3 (5^{2}) x y^{5}$ $75xy^5 = 3(5^2)xy^5$

$= 5^{2} {(y^{2})}^{2} \cdot 3 x y$ $=5^2(y^2)^2*3xy$

$= {(5 y^{2})}^{2} \cdot 3 x y$ $=(5y^2)^2*3xy$

Thus, to make it a perfect square, that is, to have all of the powers be even, we need to multiply by $3 x y$ $3xy$ :

${(5 y^{2})}^{2} \cdot 3 x y \cdot 3 x y = {(5 y^{2})}^{2} \cdot {(3 x y)}^{2} = {(15 x y^{3})}^{2}$ $(5y^2)^2*3xy*3xy = (5y^2)^2*(3xy)^2 = (15xy^3)^2$

So, using the property we mentioned initially, we can proceed:

$\frac{\sqrt{21 x^{2} y}}{\sqrt{75 x y^{5}}} = \frac{\sqrt{21 x^{2} y} \cdot \sqrt{3 x y}}{\sqrt{75 x y^{5}} \cdot \sqrt{3 x y}}$ $sqrt(21x^2y)/sqrt(75xy^5) = (sqrt(21x^2y)*sqrt(3xy))/(sqrt(75xy^5)*sqrt(3xy))$

$= \frac{\sqrt{21 x^{2} y \cdot 3 x y}}{\sqrt{75 x y^{5} \cdot 3 x y}}$ $=sqrt(21x^2y*3xy)/sqrt(75xy^5*3xy)$

$= \frac{\sqrt{63 x^{3} y^{2}}}{\sqrt{{(15 x y^{3})}^{2}}}$ $=sqrt(63x^3y^2)/sqrt((15xy^3)^2)$

$= \frac{\sqrt{63 x^{3} y^{2}}}{15 x y^{3}}$ $=sqrt(63x^3y^2)/(15xy^3)$

The denominator is now rationalized, but we can do some further simplification by pulling out any squares from the remaining square root.

$= \frac{\sqrt{(3^{2} \cdot 7) (x^{2} \cdot x) y^{2}}}{15 x y^{3}}$ $=sqrt((3^2*7)(x^2*x)y^2)/(15xy^3)$

$= \frac{\sqrt{{(3 x y)}^{2} \cdot 7 x}}{15 x y^{3}}$ $=sqrt((3xy)^2*7x)/(15xy^3)$

$= \frac{\sqrt{{(3 x y)}^{2}} \cdot \sqrt{7 x}}{15 x y^{3}}$ $=(sqrt((3xy)^2)*sqrt(7x))/(15xy^3)$

$= \frac{3 x y \sqrt{7 x}}{15 x y^{3}}$ $=(3xysqrt(7x))/(15xy^3)$

$= \frac{\sqrt{7 x}}{5 y^{2}}$ $=sqrt(7x)/(5y^2)$

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