Simultaneous Equations With Indices

The simultaneous equations
\[x^2y^3=1125\]
  (1)
\[x^3y^2-675\]
 (2)
Can most easily be solved by raising each equation to some power so that the powers of
\[x\]
( or
\[y\]
are equalised, then dividing the equations
Raise (1) to the power of 3 and (2) to the power of 2 giving
\[x^6y^9=1125^3\]
 (3)
\[x^6y^4=675^2\]
 (4)
(3) divided by (4) gives
\[\frac{x^6y^9}{x^6y^4}=\frac{1125^3}{675^2} \rightarrow y^5=3125 \rightarrow y=\sqrt[5]{3125}=5\]

Then from (1)
\[x^25^3=1125 \rightarrow x^2=\frac{1125}{5^3}=9 \rightarrow x=\sqrt{9}=3\]

Only one square root is taken since
\[x=-3\]
  does not fit equation (2)

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