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

Answer 1

Answer:

$x=\dfrac{11}{4}$

Step-by-step explanation:

Given:

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

Rewrite 9 as 3² and 27 as 3³:

$\implies (3^2)^{-\frac{1}{2}}=\dfrac{(3^3)^{\frac{1}{4}}}{3^{(x-1)}}$

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

$\implies 3^{-1}=\dfrac{3^{\frac{3}{4}}}{3^{(x-1)}}$

Multiply both sides by $3^{(x-1)}$ :

$\implies 3^{-1} \cdot 3^{(x-1)}=3^{\frac{3}{4}}$

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

$\implies 3^{(-1+x-1)}=3^{\frac{3}{4}}$

$\implies 3^{(x-2)}=3^{\frac{3}{4}}$

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

$\implies x-2=\dfrac{3}{4}$

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

$\implies 4x-8=3$

$\implies 4x=11$

$\implies x=\dfrac{11}{4}$