Question

Can someone help me find the equivalent expressions to the picture below? I’m having trouble

Answer 1

Options (1), (2), (3) and (7)

Explanation:

Given expression is
`\frac{\sqrt[3]{8^{(1)/(3)}* 3} }{3*2^{(1)/(9)}}`.

Now we will solve this expression with the help of law of exponents.

`\frac{\sqrt[3]{8^{(1)/(3)}* 3} }{3*2^{(1)/(9)}}=\frac{\sqrt[3]{(2^3)^{(1)/(3)}* 3} }{3*2^{(1)/(9)}}`

`=\frac{\sqrt[3]{2* 3} }{3*2^{(1)/(9)}}`

`=\frac{2^{(1)/(3)}* 3^{(1)/(3)}}{3* 2^{(1)/(9)}}`

`=2^{(1)/(3)}* 3^{(1)/(3)}* 2^{-(1)/(9)}* 3^(-1)`

`=2^{(1)/(3)-(1)/(9)}* 3^{(1)/(3)-1}`

`=2^{(3-1)/(9)}* 3^{(1-3)/(3)}`

`=2^{(2)/(9)}* 3^{-(2)/(3) }` [Option 2]

`2^{(2)/(9)}* 3^{-(2)/(3) }=(\sqrt[9]{2})^2* (\sqrt[3]{(1)/(3) } )^2` [Option 1]

`2^{(2)/(9)}* 3^{-(2)/(3) }=(\sqrt[9]{2})^2* (\sqrt[3]{(1)/(3) } )^2`

`=(2^2)^{(1)/(9)}* (3^2)^{-(1)/(3) }`

`=\sqrt[9]{4}* \sqrt[3]{(1)/(9) }` [Option 3]

`2^{(2)/(9)}* 3^{-(2)/(3) }=(2^2)^{(1)/(9)}* (3^(-2))^{(1)/(3) }`

`=\sqrt[9]{2^2}* \sqrt[3]{3^(-2)}` [Option 7]

Therefore, Options (1), (2), (3) and (7) are the correct options.