+112 Match each expression to its equivalent expression with the rational denominators.
Tiles
\(^1/_{\sqrt[4]{3x^3y^5}}\)
\(^3/_{\sqrt[4]{27x^{11}y^{13}}}\)
\(^2/_{\sqrt[6]{2x^7y^5}}\)
\(^4/_{\sqrt[6]{32x^5y^9}}\)
Pairs
\(^{2\sqrt[6]{2xy^3}}/_{xy^2}\to\)
\(^{\sqrt[4]{27xy^3}}/_{3xy^2}\to\)
\(^{\sqrt[4]{3xy^3}}/_{x^3y^4}\to\)
\(^{\sqrt[6]{32x^5y}}/_{x^2y}\to\)
+118670 Hi SpaceModo,
I'll do one of them.
\(\frac{{2\sqrt[6]{2xy^3}}}{_{xy^2}}\\ =\frac{{2\sqrt[6]{2xy^3}}}{\sqrt[6]{x^6y^{12}}}\\ =\frac{{2\sqrt[6]{2}}}{\sqrt[6]{x^5y^{9}}}\times\frac{2^{5/6}}{2^{5/6}}\\ =\frac{{2*2}}{\sqrt[6]{x^5y^{9}}}\times\frac{1}{(2^5)^{1/6}}\\ =\frac{{4}}{\sqrt[6]{32x^5y^{9}}}\\ \)
That is one done, now you can copy the technique and try matching up the others.
If you do not understand this one just ask and see if you can specify what line is giving you trouble :)
+112 \begin{align*}...
{^4/_{\sqrt[6]{32x^5y^9}}}&\stackrel{?}{=}{^{2\sqrt[6]{2xy^3}}/_{xy^2}}\\
{^4/_{\sqrt[6]{32x^5y^9}}}&\stackrel{?}{=}{^{2\sqrt[6]{2xy^3}}/_{\sqrt[6]{32x^5y^9}}}\\
{^4/_{\sqrt[6]{32x^5y^9}}}&\stackrel{?}{=}{^{2\sqrt[6]{2}}/_{\sqrt[6]{x^5y^9}}}\times{^{2^{^{^5}/_6}}/_{2^{^5/_6}}}\\
{^4/_{\sqrt[6]{32x^5y^9}}}&\stackrel{?}{=}{^{2\times2}/_{\sqrt[6]{x^5y^9}}}\times{^1/_{(2^5)^{^1/_6}}}\\
{^4/_{\sqrt[6]{32x^5y^9}}}&\stackrel{\checkmark}{=}{^4/_{\sqrt[6]{32x^5y^9}}}
\end{align*}
\(\begin{align*} {^4/_{\sqrt[6]{32x^5y^9}}}&\stackrel{?}{=}{^{2\sqrt[6]{2xy^3}}/_{xy^2}}\\ {^4/_{\sqrt[6]{32x^5y^9}}}&\stackrel{?}{=}{^{2\sqrt[6]{2xy^3}}/_{\sqrt[6]{32x^5y^9}}}\\ {^4/_{\sqrt[6]{32x^5y^9}}}&\stackrel{?}{=}{^{2\sqrt[6]{2}}/_{\sqrt[6]{x^5y^9}}}\times{^{2^{^5/_6}}/_{2^{^5/_6}}}\\ {^4/_{\sqrt[6]{32x^5y^9}}}&\stackrel{?}{=}{^{2\times2}/_{\sqrt[6]{x^5y^9}}}\times{^1/_{(2^5)^{^1/_6}}}\\ {^4/_{\sqrt[6]{32x^5y^9}}}&\stackrel{\checkmark}{=}{^4/_{\sqrt[6]{32x^5y^9}}} \end{align*}\)
][ |) () \/ ≡ R y ( () |\/| P |_ ≡ >< |\/| /-\ T |-| ( () |) | |\| G!
However, thank you very much for the help I needed!
+129852 4√[ 27xy^3] / [ 3xy^2] writing this in an exponential fashion, we have
[ (3^3)^(1/4) * x^(1/4) * y^(3/4) ] / [ 3xy^2] =
[( 3^3)(1/4) * x^(1/4) * y^(3/4) / [ (3^(4/4) * x^(4/4) * y^(8/4) ]
[ 3^(3/4) * x^(1/4) * y^(3/4) ] / [ 3^(4/4) * x^(4/4) * y^(8/4) ] =
{ Using a^m / a^n = a^(m - n) }
1 / [ 3^(1/4) * x^(3/4) * y^(5/4) ] write back in radical form
1 / 4√ [ 3x3y5 ]
+129852 4√[ 3xy^3 ] / [ x^3y^4 ] =
[ 3^(1/4) * x^(1/4) * y^(3/4) ] / [ x^(12/4) *y^(16/4) ] =
3^(1/4) / [ x^(11/4) * y^(13/4) ] =
Multiply top/bottom by 3^(3/4) =
[ 3^(1/4) * 3^(3/4)] / [ x^(11/4) * y^(13/4) * 3^(3/4) ] =
3 / [ x^(11/4) * y^(13/4) * 27^(1/4) ]
Write back in radical form
3 / 4√ [ 27 x11 y13 ]
