Simplify ((4^11)(5^-5))^-44^n5^m

${((4^{11}) (5^{- 5}))}^{- 4} \cdot 4^{n} \cdot 5^{m}$

Step 1

Raise $4$ to the power of $11$ .

${(4194304 \cdot 5^{- 5})}^{- 4} \cdot 4^{n} \cdot 5^{m}$

Step 2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(4194304 \cdot \frac{1}{5^{5}})}^{- 4} \cdot 4^{n} \cdot 5^{m}$

Step 3

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

${(4194304 \cdot \frac{1}{3125})}^{- 4} \cdot 4^{n} \cdot 5^{m}$

Step 4

Combine $4194304$ and $\frac{1}{3125}$ .

${(\frac{4194304}{3125})}^{- 4} \cdot 4^{n} \cdot 5^{m}$

Step 5

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

${(\frac{3125}{4194304})}^{4} \cdot 4^{n} \cdot 5^{m}$

Step 6

Simplify the expression.

Apply the product rule to $\frac{3125}{4194304}$ .

$\frac{3125^{4}}{4194304^{4}} \cdot 4^{n} \cdot 5^{m}$

Raise $3125$ to the power of $4$ .

$\frac{95367431640625}{4194304^{4}} \cdot 4^{n} \cdot 5^{m}$

Raise $4194304$ to the power of $4$ .

$\frac{95367431640625}{309485009821345000000000000} \cdot 4^{n} \cdot 5^{m}$

Step 7

Cancel the common factor of $95367431640625$ and $309485009821345000000000000$ .

Factor $1220703125$ out of $95367431640625$ .

$\frac{1220703125 (78125)}{309485009821345000000000000} \cdot 4^{n} \cdot 5^{m}$

Cancel the common factors.

Factor $1220703125$ out of $309485009821345000000000000$ .

$\frac{1220703125 \cdot 78125}{1220703125 \cdot 253530120045645824} \cdot 4^{n} \cdot 5^{m}$

Cancel the common factor.

$\frac{1220703125 \cdot 78125}{1220703125 \cdot 253530120045645824} \cdot 4^{n} \cdot 5^{m}$

Rewrite the expression.

$\frac{78125}{253530120045645824} \cdot 4^{n} \cdot 5^{m}$

Step 8

Combine $\frac{78125}{253530120045645824}$ and $4^{n}$ .

$\frac{78125 \cdot 4^{n}}{253530120045645824} \cdot 5^{m}$

Step 9

Multiply $\frac{78125 \cdot 4^{n}}{253530120045645824} \cdot 5^{m}$ .

Combine $\frac{78125 \cdot 4^{n}}{253530120045645824}$ and $5^{m}$ .

$\frac{78125 \cdot 4^{n} \cdot 5^{m}}{253530120045645824}$

Rewrite $78125$ as $5^{7}$ .

$\frac{5^{m} \cdot 5^{7} \cdot 4^{n}}{253530120045645824}$

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

$\frac{5^{m + 7} \cdot 4^{n}}{253530120045645824}$

Simplify ((4^11)(5^-5))^-4*4^n*5^m

Related Questions

Simplify ((4^11)(5^-5))^-44^n5^m