How do you multiply $\sqrt{2 x y^{3}} \cdot \sqrt{4 x^{2} y^{7}}$ $sqrt(2xy^3)*sqrt(4x^2y^7)$ ?

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How do you multiply $\sqrt{2 x y^{3}} \cdot \sqrt{4 x^{2} y^{7}}$ $sqrt(2xy^3)*sqrt(4x^2y^7)$ ?

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Answer 1 · 2015-03-24T00:51:32

You can take one big root:
$\sqrt{2 \cdot 4 x^{1 + 2} y^{3 + 7}} = \sqrt{2 \cdot 4 \cdot x^{3} y^{10}} = {(2 \cdot 2^{2} x^{3} y^{10})}^{\frac{1}{2}} =$ $sqrt(2*4x^(1+2)y^(3+7))=sqrt(2*4*x^3y^10)=(2*2^2x^3y^10)^(1/2)=$
$= 2^{\frac{1}{2}} \cdot 2^{2 \cdot \frac{1}{2}} x^{3 \cdot \frac{1}{2}} y^{10 \cdot \frac{1}{2}} =$ $=2^(1/2)*2^(2*1/2)x^(3*1/2)y^(10*1/2)=$
where you used the fact that $\sqrt{x} = x^{\frac{1}{2}}$ $sqrt(x)=x^(1/2)$
$= 2 x y^{5} \sqrt{2 x}$ $=2xy^5sqrt(2x)$

Answer 2 · 2015-03-24T01:46:45

Remember that if the exponents of two radicals are equal the arguments of the radicals can be multiplied under the same radical exponent.
That is
$\sqrt[a]{b} \times \sqrt[a]{c} = \sqrt[a]{b \times c}$ $root(a)(b) xx root(a)(c) = root(a)(b xx c)$

So
$\sqrt{2 x y^{3}} \cdot \sqrt{4 x^{2} y^{7}}$ $sqrt(2xy^3) * sqrt(4x^2y^7)$

$= \sqrt{8 x^{3} y^{10}}$ $= sqrt( 8x^3y^10)$

In this particular example some roots can be extracted:
$\sqrt{8 x^{3} y^{10}} = \sqrt{2^{2} (2) \cdot (x^{2}) (x) \cdot ({y^{5}}^{2})}$ $sqrt(8x^3y^10) = sqrt(color(red)(2)^2(2)* (color(red)(x)^2) (x) * (color(red)(y^5)^2)$

$= 2 x y^{5} \sqrt{2 x}$ $= color(red)(2xy^5) sqrt(2x)$

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How do you multiply $\sqrt{2 x y^{3}} \cdot \sqrt{4 x^{2} y^{7}}$ $sqrt(2xy^3)*sqrt(4x^2y^7)$ ?

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How do you multiply $\sqrt{2 x y^{3}} \cdot \sqrt{4 x^{2} y^{7}}$ $sqrt(2xy^3)*sqrt(4x^2y^7)$ ?