Simplify ((3^74^7)(25)^-3(5^2))/(12^75^-12^-4)

$\frac{(3^{7} \cdot 4^{7}) {(2 \cdot 5)}^{- 3} (5^{2})}{12^{7} \cdot 5^{- 1} \cdot 2^{- 4}}$

Move ${(2 \cdot 5)}^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{3^{7} \cdot 4^{7} \cdot 5^{2}}{12^{7} \cdot 5^{- 1} \cdot 2^{- 4} {(2 \cdot 5)}^{3}}$

Move $5^{- 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{3^{7} \cdot 4^{7} \cdot 5^{2} \cdot 5}{12^{7} \cdot 2^{- 4} {(2 \cdot 5)}^{3}}$

Move $2^{- 4}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{3^{7} \cdot 4^{7} \cdot 5^{2} \cdot 5 \cdot 2^{4}}{12^{7} {(2 \cdot 5)}^{3}}$

Multiply $5^{2}$ by $5$ by adding the exponents.

Move $5$ .

$\frac{3^{7} \cdot 4^{7} \cdot (5 \cdot 5^{2}) \cdot 2^{4}}{12^{7} {(2 \cdot 5)}^{3}}$

Multiply $5$ by $5^{2}$ .

Raise $5$ to the power of $1$ .

$\frac{3^{7} \cdot 4^{7} \cdot (5^{1} \cdot 5^{2}) \cdot 2^{4}}{12^{7} {(2 \cdot 5)}^{3}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3^{7} \cdot 4^{7} \cdot 5^{1 + 2} \cdot 2^{4}}{12^{7} {(2 \cdot 5)}^{3}}$

Add $1$ and $2$ .

$\frac{3^{7} \cdot 4^{7} \cdot 5^{3} \cdot 2^{4}}{12^{7} {(2 \cdot 5)}^{3}}$

Simplify the numerator.

Rewrite $4$ as $2^{2}$ .

$\frac{3^{7} \cdot (2^{4} \cdot {(2^{2})}^{7}) \cdot 5^{3}}{12^{7} {(2 \cdot 5)}^{3}}$

Multiply the exponents in ${(2^{2})}^{7}$ .

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

$\frac{3^{7} \cdot (2^{4} \cdot 2^{2 \cdot 7}) \cdot 5^{3}}{12^{7} {(2 \cdot 5)}^{3}}$

Multiply $2$ by $7$ .

$\frac{3^{7} \cdot (2^{4} \cdot 2^{14}) \cdot 5^{3}}{12^{7} {(2 \cdot 5)}^{3}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3^{7} \cdot 2^{4 + 14} \cdot 5^{3}}{12^{7} {(2 \cdot 5)}^{3}}$

Add $4$ and $14$ .

$\frac{3^{7} \cdot 2^{18} \cdot 5^{3}}{12^{7} {(2 \cdot 5)}^{3}}$

Simplify the denominator.

Apply the product rule to $2 \cdot 5$ .

$\frac{3^{7} \cdot 2^{18} \cdot 5^{3}}{12^{7} \cdot 2^{3} \cdot 5^{3}}$

Raise $12$ to the power of $7$ .

$\frac{3^{7} \cdot 2^{18} \cdot 5^{3}}{35831808 \cdot 2^{3} \cdot 5^{3}}$

Raise $2$ to the power of $3$ .

$\frac{3^{7} \cdot 2^{18} \cdot 5^{3}}{35831808 \cdot 8 \cdot 5^{3}}$

Raise $5$ to the power of $3$ .

$\frac{3^{7} \cdot 2^{18} \cdot 5^{3}}{35831808 \cdot 8 \cdot 125}$

Combine exponents.

Multiply $35831808$ by $8$ .

$\frac{3^{7} \cdot 2^{18} \cdot 5^{3}}{286654464 \cdot 125}$

Multiply $286654464$ by $125$ .

$\frac{3^{7} \cdot 2^{18} \cdot 5^{3}}{35831808000}$

Simplify the numerator.

Raise $3$ to the power of $7$ .

$\frac{2187 \cdot 2^{18} \cdot 5^{3}}{35831808000}$

Raise $2$ to the power of $18$ .

$\frac{2187 \cdot 262144 \cdot 5^{3}}{35831808000}$

Raise $5$ to the power of $3$ .

$\frac{2187 \cdot 262144 \cdot 125}{35831808000}$

Combine exponents.

Multiply $2187$ by $262144$ .

$\frac{573308928 \cdot 125}{35831808000}$

Multiply $573308928$ by $125$ .

$\frac{71663616000}{35831808000}$

Divide $71663616000$ by $35831808000$ .

$2$

Simplify ((3^7*4^7)(2*5)^-3(5^2))/(12^7*5^-1*2^-4)

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Simplify ((3^74^7)(25)^-3(5^2))/(12^75^-12^-4)