Simplify ((2e^7h^4)/(3e^2h^9))^-1((4e^2h^3)/(3eh^5))^2

${(\frac{2 e^{7} h^{4}}{3 e^{2} h^{9}})}^{- 1} {(\frac{4 e^{2} h^{3}}{3 e h^{5}})}^{2}$

Step 1

Cancel the common factor of $e^{7}$ and $e^{2}$ .

Factor $e^{2}$ out of $2 e^{7} h^{4}$ .

${(\frac{e^{2} (2 e^{5} h^{4})}{3 e^{2} h^{9}})}^{- 1} {(\frac{4 e^{2} h^{3}}{3 e h^{5}})}^{2}$

Cancel the common factors.

Factor $e^{2}$ out of $3 e^{2} h^{9}$ .

${(\frac{e^{2} (2 e^{5} h^{4})}{e^{2} (3 h^{9})})}^{- 1} {(\frac{4 e^{2} h^{3}}{3 e h^{5}})}^{2}$

Cancel the common factor.

${(\frac{e^{2} (2 e^{5} h^{4})}{e^{2} (3 h^{9})})}^{- 1} {(\frac{4 e^{2} h^{3}}{3 e h^{5}})}^{2}$

Rewrite the expression.

${(\frac{2 e^{5} h^{4}}{3 h^{9}})}^{- 1} {(\frac{4 e^{2} h^{3}}{3 e h^{5}})}^{2}$

Step 2

Cancel the common factor of $h^{4}$ and $h^{9}$ .

Factor $h^{4}$ out of $2 e^{5} h^{4}$ .

${(\frac{h^{4} (2 e^{5})}{3 h^{9}})}^{- 1} {(\frac{4 e^{2} h^{3}}{3 e h^{5}})}^{2}$

Cancel the common factors.

Factor $h^{4}$ out of $3 h^{9}$ .

${(\frac{h^{4} (2 e^{5})}{h^{4} (3 h^{5})})}^{- 1} {(\frac{4 e^{2} h^{3}}{3 e h^{5}})}^{2}$

Cancel the common factor.

${(\frac{h^{4} (2 e^{5})}{h^{4} (3 h^{5})})}^{- 1} {(\frac{4 e^{2} h^{3}}{3 e h^{5}})}^{2}$

Rewrite the expression.

${(\frac{2 e^{5}}{3 h^{5}})}^{- 1} {(\frac{4 e^{2} h^{3}}{3 e h^{5}})}^{2}$

Step 3

Change the sign of the exponent by rewriting the base as its reciprocal.

$\frac{3 h^{5}}{2 e^{5}} {(\frac{4 e^{2} h^{3}}{3 e h^{5}})}^{2}$

Step 4

Cancel the common factor of $e^{2}$ and $e$ .

Factor $e$ out of $4 e^{2} h^{3}$ .

$\frac{3 h^{5}}{2 e^{5}} {(\frac{e (4 e h^{3})}{3 e h^{5}})}^{2}$

Cancel the common factors.

Factor $e$ out of $3 e h^{5}$ .

$\frac{3 h^{5}}{2 e^{5}} {(\frac{e (4 e h^{3})}{e (3 h^{5})})}^{2}$

Cancel the common factor.

$\frac{3 h^{5}}{2 e^{5}} {(\frac{e (4 e h^{3})}{e (3 h^{5})})}^{2}$

Rewrite the expression.

$\frac{3 h^{5}}{2 e^{5}} {(\frac{4 e h^{3}}{3 h^{5}})}^{2}$

Step 5

Cancel the common factor of $h^{3}$ and $h^{5}$ .

Factor $h^{3}$ out of $4 e h^{3}$ .

$\frac{3 h^{5}}{2 e^{5}} {(\frac{h^{3} (4 e)}{3 h^{5}})}^{2}$

Cancel the common factors.

Factor $h^{3}$ out of $3 h^{5}$ .

$\frac{3 h^{5}}{2 e^{5}} {(\frac{h^{3} (4 e)}{h^{3} (3 h^{2})})}^{2}$

Cancel the common factor.

$\frac{3 h^{5}}{2 e^{5}} {(\frac{h^{3} (4 e)}{h^{3} (3 h^{2})})}^{2}$

Rewrite the expression.

$\frac{3 h^{5}}{2 e^{5}} {(\frac{4 e}{3 h^{2}})}^{2}$

Step 6

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{4 e}{3 h^{2}}$ .

$\frac{3 h^{5}}{2 e^{5}} \cdot \frac{{(4 e)}^{2}}{{(3 h^{2})}^{2}}$

Apply the product rule to $4 e$ .

$\frac{3 h^{5}}{2 e^{5}} \cdot \frac{4^{2} e^{2}}{{(3 h^{2})}^{2}}$

Apply the product rule to $3 h^{2}$ .

$\frac{3 h^{5}}{2 e^{5}} \cdot \frac{4^{2} e^{2}}{3^{2} {(h^{2})}^{2}}$

Step 7

Combine.

$\frac{3 h^{5} (4^{2} e^{2})}{2 e^{5} (3^{2} {(h^{2})}^{2})}$

Step 8

Cancel the common factor of $3$ and $3^{2}$ .

Factor $3$ out of $3 h^{5} (4^{2} e^{2})$ .

$\frac{3 (h^{5} (4^{2} e^{2}))}{2 e^{5} (3^{2} {(h^{2})}^{2})}$

Cancel the common factors.

Factor $3$ out of $2 e^{5} (3^{2} {(h^{2})}^{2})$ .

$\frac{3 (h^{5} (4^{2} e^{2}))}{3 (2 e^{5} (3 {(h^{2})}^{2}))}$

Cancel the common factor.

$\frac{3 (h^{5} (4^{2} e^{2}))}{3 (2 e^{5} (3 {(h^{2})}^{2}))}$

Rewrite the expression.

$\frac{h^{5} (4^{2} e^{2})}{2 e^{5} (3 {(h^{2})}^{2})}$

Step 9

Cancel the common factor of $e^{2}$ and $e^{5}$ .

Factor $e^{2}$ out of $h^{5} (4^{2} e^{2})$ .

$\frac{e^{2} (h^{5} \cdot (4^{2}))}{2 e^{5} (3 {(h^{2})}^{2})}$

Cancel the common factors.

Factor $e^{2}$ out of $2 e^{5} (3 {(h^{2})}^{2})$ .

$\frac{e^{2} (h^{5} \cdot (4^{2}))}{e^{2} (2 e^{3} (3 {(h^{2})}^{2}))}$

Cancel the common factor.

$\frac{e^{2} (h^{5} \cdot (4^{2}))}{e^{2} (2 e^{3} (3 {(h^{2})}^{2}))}$

Rewrite the expression.

$\frac{h^{5} \cdot (4^{2})}{2 e^{3} (3 {(h^{2})}^{2})}$

Step 10

Simplify the expression.

Raise $4$ to the power of $2$ .

$\frac{h^{5} \cdot 16}{2 e^{3} \cdot 3 {(h^{2})}^{2}}$

Multiply $3$ by $2$ .

$\frac{h^{5} \cdot 16}{6 e^{3} {(h^{2})}^{2}}$

Multiply the exponents in ${(h^{2})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{h^{5} \cdot 16}{6 e^{3} h^{2 \cdot 2}}$

Multiply $2$ by $2$ .

$\frac{h^{5} \cdot 16}{6 e^{3} h^{4}}$

Step 11

Cancel the common factor of $h^{5}$ and $h^{4}$ .

Factor $h^{4}$ out of $h^{5} \cdot 16$ .

$\frac{h^{4} (h \cdot 16)}{6 e^{3} h^{4}}$

Cancel the common factors.

Factor $h^{4}$ out of $6 e^{3} h^{4}$ .

$\frac{h^{4} (h \cdot 16)}{h^{4} (6 e^{3})}$

Cancel the common factor.

$\frac{h^{4} (h \cdot 16)}{h^{4} (6 e^{3})}$

Rewrite the expression.

$\frac{h \cdot 16}{6 e^{3}}$

Step 12

Cancel the common factor of $16$ and $6$ .

Factor $2$ out of $h \cdot 16$ .

$\frac{2 (h \cdot 8)}{6 e^{3}}$

Cancel the common factors.

Factor $2$ out of $6 e^{3}$ .

$\frac{2 (h \cdot 8)}{2 (3 e^{3})}$

Cancel the common factor.

$\frac{2 (h \cdot 8)}{2 (3 e^{3})}$

Rewrite the expression.

$\frac{h \cdot 8}{3 e^{3}}$

Step 13

Move $8$ to the left of $h$ .

$\frac{8 h}{3 e^{3}}$

Simplify ((2e^7h^4)/(3e^2h^9))^-1((4e^2h^3)/(3eh^5))^2

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