\{ \frac { ( \sqrt { 3 } ) * 3 ^ { - 2 } } { ( \sqrt { 5 } ) ^ { 2 } } \} ^ { \frac { 1 } { 2 } }solve this equation ​

Question

`\{ \frac { ( \sqrt { 3 } ) * 3 ^ { - 2 } } { ( \sqrt { 5 } ) ^ { 2 } } \} ^ { \frac { 1 } { 2 } }`solve this equation

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Answer 1

Explanation:

Exponent law:

`\sf \bf a^m * a^n = a^(m+n)\\\\ (a^m)^n = a^(m*n)`

`\sf a^(-m)=(1)/(a^m)`

First convert radical form to exponent form and then apply exponent law.

`\sf √(3)=3^{(1)/(2)}\\\\√(5)=5^{(1)/(2)}`

`\sf \left(((√(3)*3^(-2))/((√(5))^2)\right)^{(1)/(2)}= \left(\frac{3^{(1)/(2)}*3^(-2)}{(5^{(1)/(2)})^2} \right )^{(1)/(2)}`

`= \left(\frac{3^{(1)/(2)-2}}{5^{(1)/(2)*2}}\right)^{(1)/(2)}\\\\=\left(\frac{3^{(1-4)/(2)}}{5}\right)^{(1)/(2)}\\\\=\left(\frac{3^{(-3)/(2)}}{5}\right)^{(1)/(2)}\\\\=\frac{3^{(-3)/(2)*{(1)/(2)}}}{5^{(1)/(2)}}\\\\ =\frac{3^{{(-3)/(4)}}}{5^{(1)/(2)}}`