Question

9 to the power of - half is equal to (as a fraction) 27 to the power of a quarter over 3 to the power of x-1

whats x?
9^(-1/2) = ((27^(1/4))/(3^(x-1)))

Answer 1

`x=(11)/(4)`

Explanation:

Given:

`9^{-(1)/(2)}=\frac{27^{(1)/(4)}}{3^((x-1))}`

Rewrite 9 as 3² and 27 as 3³:

`\implies (3^2)^{-(1)/(2)}=\frac{(3^3)^{(1)/(4)}}{3^((x-1))}`

`\textsf{Apply exponent rule} \quad (a^b)^c=a^(bc):`

`\implies 3^(-1)=\frac{3^{(3)/(4)}}{3^((x-1))}`

Multiply both sides by
`3^((x-1))` :

`\implies 3^(-1) \cdot 3^((x-1))=3^{(3)/(4)}`

`\textsf{Apply exponent rule} \quad a^b \cdot a^c=a^(b+c):`

`\implies 3^((-1+x-1))=3^{(3)/(4)}`

`\implies 3^((x-2))=3^{(3)/(4)}`

`\textsf{Apply exponent rule} \quad a^(f(x))=a^(g(x)) \implies f(x)=g(x):`

`\implies x-2=(3)/(4)`

`\implies 4(x-2)=3`

`\implies 4x-8=3`

`\implies 4x=11`

`\implies x=(11)/(4)`