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

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

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Answer 1 · 2016-12-11T04:35:46

$15 x^{9} y^{11}$ $15x^9y^11$

Explanation:

Recall : $\sqrt{x} = x^{\frac{1}{2}}, and \sqrt{x^{2}} = {(x^{2})}^{\frac{1}{2}} = x^{2 \cdot \frac{1}{2}} = x$ $sqrtx=x^(1/2), and sqrt(x^2)=(x^2)^(1/2)=x^(2*1/2)=x$ ,

$\sqrt{225 x^{18} y^{22}} = \sqrt{15 \cdot 15 \cdot x^{18} \cdot y^{22}} = {(15^{2} \cdot x^{18} \cdot y^{22})}^{\frac{1}{2}}$ $sqrt(225x^18y^22)=sqrt(15*15*x^18*y^22)=(15^2*x^18*y^22)^(1/2)$
$= 15^{2 \cdot \frac{1}{2}} \cdot x^{18 \cdot \frac{1}{2}} \cdot y^{22 \cdot \frac{1}{2}}$ $=15^(2*1/2)*x^(18*1/2)*y^(22*1/2)$
$= 15 \cdot x^{9} \cdot y^{11}$ $=15*x^9*y^11$

Answer 2 · 2016-12-11T09:07:09

$\sqrt{225 x^{18} y^{22}} = ∣ ∣ 15 x^{9} y^{11} ∣ ∣$ $sqrt(225x^18y^22) = abs(15x^9y^11)$

Explanation:

Note that:

$225 x^{18} y^{22} = 15^{2} \cdot {(x^{9})}^{2} \cdot {(y^{11})}^{2} = {(15 x^{9} y^{11})}^{2}$ $225x^18y^22 = 15^2*(x^9)^2*(y^11)^2 = (15x^9y^11)^2$

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

We also find:

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

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

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

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

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

$\sqrt{225 x^{18} y^{22}} = ∣ ∣ 15 x^{9} y^{11} ∣ ∣$ $sqrt(225x^18y^22) = abs(15x^9y^11)$

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

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