# How do you solve 1/ sqrt 8 = 4^(m – 2)?

Feb 14, 2016

The answer is $m = 1 \frac{1}{4}$

#### Explanation:

When solving exponential equations (or inequalities) first you have to find a suitable common base. In this case it would be $2$ because $8 = {2}^{3}$ and $4 = {2}^{2}$.

Now you have to write the equation using calculated base:

$\frac{1}{\sqrt{{2}^{3}}} = {2}^{2 \cdot \left(m - 2\right)}$

Now you can use the property of powers which says that $\sqrt[n]{a} = {a}^{\frac{1}{n}}$

$\frac{1}{2} ^ \left(\frac{3}{2}\right) = {2}^{2 m - 4}$

Next property to use is: $\frac{1}{{a}^{x}} = {a}^{- x}$

${2}^{- \frac{3}{2}} = {2}^{2 m - 4}$

Now since we have the equality of 2 powers with equal base we can write it as the equality of exponents:

$- \frac{3}{2} = 2 m - 4$

$2 m = 4 - \frac{3}{2}$

$2 m = 2 \frac{1}{2}$

$2 m = \frac{5}{2}$

$m = \frac{5}{4} = 1 \frac{1}{4}$