If a and b are distinct primes such that [3] a^6b^ -4 =a^xb^ 2y ,find x and y.

[3] a^6b^ -4 =a^xb^ 2y (a^6b ^ -4 )^1 3 = a^x ^ 2y ^ 6 1 3 ^ -4 3 =a^x ^ 2y ^2 ^ -4 3 =a^x ^ 2y 2=x and -4 3 =2y =2 and y=-2 3

If a and b are distinct primes such that [3] a^6b^ -4 =a^xb^ 2y ,…

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Question

If a and b are distinct primes such that $\sqrt[3]{\text{a}^6\text{b}^{-4}}=\text{a}^\text{x}\text{b}^{2\text{y}},$find x and y.

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Answer

$\sqrt[3]{\text{a}^6\text{b}^{-4}}=\text{a}^\text{x}\text{b}^{2\text{y}}$
$\Rightarrow(\text{a}^6\text{b}^{-4})^\frac{1}{3}=\text{a}^\text{x}\times\text{b}^{2\text{y}}$
$\Rightarrow\text{a}^{6\times\frac{1}{3}}\times\text{b}^{-\frac{4}{3}}=\text{a}^\text{x}\times\text{b}^{2\text{y}}$
$\Rightarrow\text{a}^2\times\text{b}^{-\frac{4}{3}}=\text{a}^\text{x}\times\text{b}^{2\text{y}}$
$\Rightarrow2=\text{x}$ and $-\frac{4}{3}=2\text{y}$
$\Rightarrow\text{x}=2$ and $\text{y}=-\frac{2}{3}$

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