# How do you simplify 2^(5/2) - 2^(3/2)?

Apr 9, 2015

In general
${b}^{p + q} = {b}^{p} \cdot {b}^{q}$
so
${2}^{\frac{5}{2}}$ can be re-written as ${2}^{\frac{3}{2}} \cdot {2}^{\frac{2}{2}} = {2}^{\frac{3}{2}} \cdot 2$

${2}^{\frac{5}{2}} - {2}^{\frac{3}{2}}$

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

and since
${2}^{\frac{3}{2}} = {\left({2}^{3}\right)}^{\frac{1}{2}} = 2 \sqrt{2}$

${2}^{\frac{5}{2}} - {2}^{\frac{3}{2}} = 2 \sqrt{2}$

Jul 17, 2017

$2 \sqrt{2}$

#### Explanation:

Do not be tempted to simplify the indices. These are separate terms.

However, we can factorise and take out a common factor of ${2}^{\frac{3}{2}}$.

When you are dividing and the bases are the same, subtract the indices.

${2}^{\frac{5}{2}} - {2}^{\frac{3}{2}} = {2}^{\frac{3}{2}} \left({2}^{\frac{2}{2}} - 1\right)$

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

${2}^{\frac{3}{2}}$ can also be written as: $\text{ } {2}^{\frac{2}{2} + \frac{1}{2}} = 2 \times {2}^{\frac{1}{2}}$

$= 2 \sqrt{2}$

Another approach to simplify this is:

${2}^{\frac{3}{2}} = \sqrt{{2}^{3}} = \sqrt{8} = \sqrt{4 \times 2}$

$= 2 \sqrt{2}$