Simplify (2x^2y^-2m^3)^4(2x^4y^5m^8)

${(2 x^{2} y^{- 2} m^{3})}^{4} (2 x^{4} y^{5} m^{8})$

Step 1

Rewrite using the commutative property of multiplication.

$2 {(2 x^{2} y^{- 2} m^{3})}^{4} x^{4} y^{5} m^{8}$

Step 2

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

$2 {(2 x^{2} \frac{1}{y^{2}} m^{3})}^{4} x^{4} y^{5} m^{8}$

Step 3

Multiply $2 x^{2} \frac{1}{y^{2}}$ .

Combine $2$ and $\frac{1}{y^{2}}$ .

$2 {(x^{2} \frac{2}{y^{2}} m^{3})}^{4} x^{4} y^{5} m^{8}$

Combine $x^{2}$ and $\frac{2}{y^{2}}$ .

$2 {(\frac{x^{2} \cdot 2}{y^{2}} m^{3})}^{4} x^{4} y^{5} m^{8}$

Step 4

Move $2$ to the left of $x^{2}$ .

$2 {(\frac{2 \cdot x^{2}}{y^{2}} m^{3})}^{4} x^{4} y^{5} m^{8}$

Step 5

Combine $\frac{2 x^{2}}{y^{2}}$ and $m^{3}$ .

$2 {(\frac{2 x^{2} m^{3}}{y^{2}})}^{4} x^{4} y^{5} m^{8}$

Step 6

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

Apply the product rule to $\frac{2 x^{2} m^{3}}{y^{2}}$ .

$2 \frac{{(2 x^{2} m^{3})}^{4}}{{(y^{2})}^{4}} x^{4} y^{5} m^{8}$

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

$2 \frac{{(2 x^{2})}^{4} {(m^{3})}^{4}}{{(y^{2})}^{4}} x^{4} y^{5} m^{8}$

Apply the product rule to $2 x^{2}$ .

$2 \frac{2^{4} {(x^{2})}^{4} {(m^{3})}^{4}}{{(y^{2})}^{4}} x^{4} y^{5} m^{8}$

Step 7

Simplify the numerator.

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

$2 \frac{16 {(x^{2})}^{4} {(m^{3})}^{4}}{{(y^{2})}^{4}} x^{4} y^{5} m^{8}$

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

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

$2 \frac{16 x^{2 \cdot 4} {(m^{3})}^{4}}{{(y^{2})}^{4}} x^{4} y^{5} m^{8}$

Multiply $2$ by $4$ .

$2 \frac{16 x^{8} {(m^{3})}^{4}}{{(y^{2})}^{4}} x^{4} y^{5} m^{8}$

Multiply the exponents in ${(m^{3})}^{4}$ .

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

$2 \frac{16 x^{8} m^{3 \cdot 4}}{{(y^{2})}^{4}} x^{4} y^{5} m^{8}$

Multiply $3$ by $4$ .

$2 \frac{16 x^{8} m^{12}}{{(y^{2})}^{4}} x^{4} y^{5} m^{8}$

Step 8

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

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

$2 \frac{16 x^{8} m^{12}}{y^{2 \cdot 4}} x^{4} y^{5} m^{8}$

Multiply $2$ by $4$ .

$2 \frac{16 x^{8} m^{12}}{y^{8}} x^{4} y^{5} m^{8}$

Step 9

Multiply $2 \frac{16 x^{8} m^{12}}{y^{8}}$ .

Combine $2$ and $\frac{16 x^{8} m^{12}}{y^{8}}$ .

$\frac{2 (16 x^{8} m^{12})}{y^{8}} x^{4} y^{5} m^{8}$

Multiply $16$ by $2$ .

$\frac{32 (x^{8} m^{12})}{y^{8}} x^{4} y^{5} m^{8}$

Step 10

Multiply $\frac{32 x^{8} m^{12}}{y^{8}} x^{4}$ .

Combine $\frac{32 x^{8} m^{12}}{y^{8}}$ and $x^{4}$ .

$\frac{32 x^{8} m^{12} x^{4}}{y^{8}} y^{5} m^{8}$

Multiply $x^{8}$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

$\frac{32 (x^{4} x^{8}) m^{12}}{y^{8}} y^{5} m^{8}$

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

$\frac{32 x^{4 + 8} m^{12}}{y^{8}} y^{5} m^{8}$

Add $4$ and $8$ .

$\frac{32 x^{12} m^{12}}{y^{8}} y^{5} m^{8}$

Step 11

Cancel the common factor of $y^{5}$ .

Factor $y^{5}$ out of $y^{8}$ .

$\frac{32 x^{12} m^{12}}{y^{5} y^{3}} y^{5} m^{8}$

Cancel the common factor.

$\frac{32 x^{12} m^{12}}{y^{5} y^{3}} y^{5} m^{8}$

Rewrite the expression.

$\frac{32 x^{12} m^{12}}{y^{3}} m^{8}$

Step 12

Multiply $\frac{32 x^{12} m^{12}}{y^{3}} m^{8}$ .

Combine $\frac{32 x^{12} m^{12}}{y^{3}}$ and $m^{8}$ .

$\frac{32 x^{12} m^{12} m^{8}}{y^{3}}$

Multiply $m^{12}$ by $m^{8}$ by adding the exponents.

Move $m^{8}$ .

$\frac{32 x^{12} (m^{8} m^{12})}{y^{3}}$

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

$\frac{32 x^{12} m^{8 + 12}}{y^{3}}$

Add $8$ and $12$ .

$\frac{32 x^{12} m^{20}}{y^{3}}$

Simplify (2x^2y^-2m^3)^4(2x^4y^5m^8)

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