Simplify (m^4n^6)^4(m^3n^2p^5)^6

${(m^{4} n^{6})}^{4} {(m^{3} n^{2} p^{5})}^{6}$

Step 1

Apply the product rule to $m^{4} n^{6}$ .

${(m^{4})}^{4} {(n^{6})}^{4} {(m^{3} n^{2} p^{5})}^{6}$

Step 2

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

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

$m^{4 \cdot 4} {(n^{6})}^{4} {(m^{3} n^{2} p^{5})}^{6}$

Multiply $4$ by $4$ .

$m^{16} {(n^{6})}^{4} {(m^{3} n^{2} p^{5})}^{6}$

Step 3

Multiply the exponents in ${(n^{6})}^{4}$ .

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

$m^{16} n^{6 \cdot 4} {(m^{3} n^{2} p^{5})}^{6}$

Multiply $6$ by $4$ .

$m^{16} n^{24} {(m^{3} n^{2} p^{5})}^{6}$

Step 4

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

Apply the product rule to $m^{3} n^{2} p^{5}$ .

$m^{16} n^{24} ({(m^{3} n^{2})}^{6} {(p^{5})}^{6})$

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

$m^{16} n^{24} ({(m^{3})}^{6} {(n^{2})}^{6} {(p^{5})}^{6})$

Step 5

Multiply the exponents in ${(m^{3})}^{6}$ .

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

$m^{16} n^{24} (m^{3 \cdot 6} {(n^{2})}^{6} {(p^{5})}^{6})$

Multiply $3$ by $6$ .

$m^{16} n^{24} (m^{18} {(n^{2})}^{6} {(p^{5})}^{6})$

Step 6

Multiply $m^{16}$ by $m^{18}$ by adding the exponents.

Move $m^{18}$ .

$m^{18} m^{16} n^{24} ({(n^{2})}^{6} {(p^{5})}^{6})$

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

$m^{18 + 16} n^{24} ({(n^{2})}^{6} {(p^{5})}^{6})$

Add $18$ and $16$ .

$m^{34} n^{24} ({(n^{2})}^{6} {(p^{5})}^{6})$

Step 7

Multiply the exponents in ${(n^{2})}^{6}$ .

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

$m^{34} n^{24} (n^{2 \cdot 6} {(p^{5})}^{6})$

Multiply $2$ by $6$ .

$m^{34} n^{24} (n^{12} {(p^{5})}^{6})$

Step 8

Multiply $n^{24}$ by $n^{12}$ by adding the exponents.

Move $n^{12}$ .

$m^{34} (n^{12} n^{24}) {(p^{5})}^{6}$

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

$m^{34} n^{12 + 24} {(p^{5})}^{6}$

Add $12$ and $24$ .

$m^{34} n^{36} {(p^{5})}^{6}$

Step 9

Multiply the exponents in ${(p^{5})}^{6}$ .

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

$m^{34} n^{36} p^{5 \cdot 6}$

Multiply $5$ by $6$ .

$m^{34} n^{36} p^{30}$

Simplify (m^4n^6)^4(m^3n^2p^5)^6

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