Question

Evaluate the question

Answer 1

we conclude that:

`\frac{2^{-(4)/(3)}}{54^{-(4)/(3)}}=81`

Explanation:

Given the expression

`\frac{2^{-(4)/(3)}}{54^{-(4)/(3)}}`

`\mathrm{Apply\:exponent\:rule}:\quad (x^a)/(x^b)\:=\:x^(a-b)`

`\frac{2^{-(4)/(3)}}{54^{-(4)/(3)}}=\left((2)/(54)\right)^{-(4)/(3)}`

`\mathrm{Apply\:exponent\:rule}:\quad \:a^(-b)=(1)/(a^b)`

`=\frac{1}{\left((2)/(54)\right)^{(4)/(3)}}`

`=\left((1)/(27)\right)^{-(4)/(3)}`

`\mathrm{Apply\:exponent\:rule}:\quad \left((a)/(b)\right)^c=(a^c)/(b^c)`

`=\frac{1}{\frac{1^{(4)/(3)}}{27^{(4)/(3)}}}`

`=\frac{1}{\frac{1^{(4)/(3)}}{81}}` ∵
`27^{(4)/(3)}=81`

`\mathrm{Apply\:the\:fraction\:rule}:\quad (1)/((b)/(c))=(c)/(b)`

`=\frac{81}{1^{(1)/(3)}}`

`\mathrm{Apply\:rule}\:1^a=1`

`=(81)/(1)`

`=81`

Therefore, we conclude that:

`\frac{2^{-(4)/(3)}}{54^{-(4)/(3)}}=81`