Simplify (5k^2m)^3((4(km^4))^2)^2

${(5 k^{2} m)}^{3} {({(4 (k m^{4}))}^{2})}^{2}$

Step 1

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

Apply the product rule to $5 k^{2} m$ .

${(5 k^{2})}^{3} m^{3} {({(4 (k m^{4}))}^{2})}^{2}$

Apply the product rule to $5 k^{2}$ .

$5^{3} {(k^{2})}^{3} m^{3} {({(4 (k m^{4}))}^{2})}^{2}$

Step 2

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

$125 {(k^{2})}^{3} m^{3} {({(4 (k m^{4}))}^{2})}^{2}$

Step 3

Multiply the exponents in ${(k^{2})}^{3}$ .

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

$125 k^{2 \cdot 3} m^{3} {({(4 (k m^{4}))}^{2})}^{2}$

Multiply $2$ by $3$ .

$125 k^{6} m^{3} {({(4 (k m^{4}))}^{2})}^{2}$

Step 4

Multiply the exponents in ${({(4 (k m^{4}))}^{2})}^{2}$ .

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

$125 k^{6} m^{3} {(4 (k m^{4}))}^{2 \cdot 2}$

Multiply $2$ by $2$ .

$125 k^{6} m^{3} {(4 (k m^{4}))}^{4}$

Step 5

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

Apply the product rule to $4 k m^{4}$ .

$125 k^{6} m^{3} ({(4 k)}^{4} {(m^{4})}^{4})$

Apply the product rule to $4 k$ .

$125 k^{6} m^{3} (4^{4} k^{4} {(m^{4})}^{4})$

Step 6

Multiply $k^{6}$ by $k^{4}$ by adding the exponents.

Move $k^{4}$ .

$125 (k^{4} k^{6}) m^{3} (4^{4} {(m^{4})}^{4})$

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

$125 k^{4 + 6} m^{3} (4^{4} {(m^{4})}^{4})$

Add $4$ and $6$ .

$125 k^{10} m^{3} (4^{4} {(m^{4})}^{4})$

Step 7

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

$125 k^{10} m^{3} (256 {(m^{4})}^{4})$

Step 8

Multiply the exponents in ${(m^{4})}^{4}$ .

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

$125 k^{10} m^{3} (256 m^{4 \cdot 4})$

Multiply $4$ by $4$ .

$125 k^{10} m^{3} (256 m^{16})$

Step 9

Multiply $m^{3}$ by $m^{16}$ by adding the exponents.

Move $m^{16}$ .

$125 k^{10} (m^{16} m^{3}) \cdot 256$

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

$125 k^{10} m^{16 + 3} \cdot 256$

Add $16$ and $3$ .

$125 k^{10} m^{19} \cdot 256$

Step 10

Multiply $256$ by $125$ .

$32000 k^{10} m^{19}$

Simplify (5k^2m)^3((4(km^4))^2)^2

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