# How do you simplify ((c^2d^2 )/ (cd^3)) * (d^2 / c^2)^3 leaving only positive exponents?

Mar 15, 2018

((c^2d^2)/(cd^3))*((d^2)/(c^2))^3=color(blue)(d^5/c^5

#### Explanation:

Simplify.

$\left(\frac{{c}^{2} {d}^{2}}{c {d}^{3}}\right) \cdot {\left(\frac{{d}^{2}}{{c}^{2}}\right)}^{3}$

Apply power rule of exponents: ${\left({a}^{m}\right)}^{n} = {a}^{m \cdot n}$

$\left(\frac{{c}^{2} {d}^{2}}{c {d}^{3}}\right) \cdot \left({d}^{\left(2 \cdot 3\right)} / \left({c}^{\left(2 \cdot 3\right)}\right)\right)$

Simplify.

$\left(\frac{{c}^{2} {d}^{2}}{c {d}^{3}}\right) \cdot \left(\frac{{d}^{6}}{{c}^{6}}\right)$

Remove parentheses.

$\frac{{c}^{2} {d}^{2}}{c {d}^{3}} \cdot \frac{{d}^{6}}{{c}^{6}}$

Apply product rule of exponents: ${a}^{m} {a}^{n} = {a}^{m + n}$

No exponent is understood to be an exponent of $1$.

$\frac{{c}^{2} {d}^{\left(2 + 6\right)}}{{c}^{\left(1 + 6\right)} {d}^{3}}$

Simplify.

$\frac{{c}^{2} {d}^{8}}{{c}^{7} {d}^{3}}$

Apply quotient rule of exponents: $\frac{{a}^{m}}{{a}^{n}} = {a}^{m - n}$

${c}^{\left(2 - 7\right)} {d}^{\left(8 - 3\right)}$

Simplify.

${c}^{- 5} {d}^{5}$

Apply negative exponent rule: ${a}^{- m} = \frac{1}{a} ^ m$

${d}^{5} / {c}^{5}$