# How do you solve (1/4)^(2x)=(1/32)^(3x+1)?

May 26, 2016

$x = - \frac{5}{11}$

#### Explanation:

${\left(\frac{1}{4}\right)}^{2 x} = {\left(\frac{1}{32}\right)}^{3 x + 1} \equiv \frac{1}{4} ^ \left\{2 x\right\} = \frac{1}{32} ^ \left\{3 x + 1\right\}$
$\frac{1}{4} ^ \left\{2 x\right\} = \frac{1}{32} ^ \left\{3 x + 1\right\} \equiv {\left(4\right)}^{2 x} = {\left(32\right)}^{3 x + 1}$
${\left(4\right)}^{2 x} = {\left(32\right)}^{3 x + 1} \equiv {\left({2}^{2}\right)}^{2 x} = {\left({2}^{5}\right)}^{3 x + 1}$
${\left({2}^{2}\right)}^{2 x} = {\left({2}^{5}\right)}^{3 x + 1} \equiv {2}^{2 \times 2 x} = {2}^{5 \times \left(3 x + 1\right)}$
${2}^{2 \times 2 x} = {2}^{5 \times \left(3 x + 1\right)} \equiv 4 x = 15 x + 5$
solving for $x$ we get $x = - \frac{5}{11}$