Simplify ((2a^0b^-4)/(b^3))^4*((a^3b^-2)/(3b^4a^-4))^3

${(\frac{2 a^{0} b^{- 4}}{b^{3}})}^{4} \cdot {(\frac{a^{3} b^{- 2}}{3 b^{4} a^{- 4}})}^{3}$

Step 1

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

${(\frac{2}{b^{3} b^{4}})}^{4} \cdot {(\frac{a^{3} b^{- 2}}{3 b^{4} a^{- 4}})}^{3}$

Step 2

Multiply $b^{3}$ by $b^{4}$ by adding the exponents.

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

${(\frac{2}{b^{3 + 4}})}^{4} \cdot {(\frac{a^{3} b^{- 2}}{3 b^{4} a^{- 4}})}^{3}$

Add $3$ and $4$ .

${(\frac{2}{b^{7}})}^{4} \cdot {(\frac{a^{3} b^{- 2}}{3 b^{4} a^{- 4}})}^{3}$

Step 3

Apply the product rule to $\frac{2}{b^{7}}$ .

$\frac{2^{4}}{{(b^{7})}^{4}} \cdot {(\frac{a^{3} b^{- 2}}{3 b^{4} a^{- 4}})}^{3}$

Step 4

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

$\frac{16}{{(b^{7})}^{4}} \cdot {(\frac{a^{3} b^{- 2}}{3 b^{4} a^{- 4}})}^{3}$

Step 5

Multiply the exponents in ${(b^{7})}^{4}$ .

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

$\frac{16}{b^{7 \cdot 4}} \cdot {(\frac{a^{3} b^{- 2}}{3 b^{4} a^{- 4}})}^{3}$

Multiply $7$ by $4$ .

$\frac{16}{b^{28}} \cdot {(\frac{a^{3} b^{- 2}}{3 b^{4} a^{- 4}})}^{3}$

Step 6

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

$\frac{16}{b^{28}} \cdot {(\frac{a^{3}}{3 b^{4} a^{- 4} b^{2}})}^{3}$

Step 7

Multiply $b^{4}$ by $b^{2}$ by adding the exponents.

Move $b^{2}$ .

$\frac{16}{b^{28}} \cdot {(\frac{a^{3}}{3 (b^{2} b^{4}) a^{- 4}})}^{3}$

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

$\frac{16}{b^{28}} \cdot {(\frac{a^{3}}{3 b^{2 + 4} a^{- 4}})}^{3}$

Add $2$ and $4$ .

$\frac{16}{b^{28}} \cdot {(\frac{a^{3}}{3 b^{6} a^{- 4}})}^{3}$

Step 8

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

$\frac{16}{b^{28}} \cdot {(\frac{a^{3} a^{4}}{3 b^{6}})}^{3}$

Step 9

Multiply $a^{3}$ by $a^{4}$ by adding the exponents.

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

$\frac{16}{b^{28}} \cdot {(\frac{a^{3 + 4}}{3 b^{6}})}^{3}$

Add $3$ and $4$ .

$\frac{16}{b^{28}} \cdot {(\frac{a^{7}}{3 b^{6}})}^{3}$

Step 10

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

Apply the product rule to $\frac{a^{7}}{3 b^{6}}$ .

$\frac{16}{b^{28}} \cdot \frac{{(a^{7})}^{3}}{{(3 b^{6})}^{3}}$

Apply the product rule to $3 b^{6}$ .

$\frac{16}{b^{28}} \cdot \frac{{(a^{7})}^{3}}{3^{3} {(b^{6})}^{3}}$

Step 11

Combine fractions.

Combine.

$\frac{16 {(a^{7})}^{3}}{b^{28} (3^{3} {(b^{6})}^{3})}$

Multiply the exponents in ${(a^{7})}^{3}$ .

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

$\frac{16 a^{7 \cdot 3}}{b^{28} (3^{3} {(b^{6})}^{3})}$

Multiply $7$ by $3$ .

$\frac{16 a^{21}}{b^{28} (3^{3} {(b^{6})}^{3})}$

Step 12

Simplify the denominator.

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

$\frac{16 a^{21}}{b^{28} \cdot 27 {(b^{6})}^{3}}$

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

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

$\frac{16 a^{21}}{b^{28} \cdot 27 b^{6 \cdot 3}}$

Multiply $6$ by $3$ .

$\frac{16 a^{21}}{b^{28} \cdot 27 b^{18}}$

Multiply $b^{28}$ by $b^{18}$ by adding the exponents.

Move $b^{18}$ .

$\frac{16 a^{21}}{b^{18} b^{28} \cdot 27}$

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

$\frac{16 a^{21}}{b^{18 + 28} \cdot 27}$

Add $18$ and $28$ .

$\frac{16 a^{21}}{b^{46} \cdot 27}$

Step 13

Move $27$ to the left of $b^{46}$ .

$\frac{16 a^{21}}{27 b^{46}}$

Simplify ((2a^0b^-4)/(b^3))^4*((a^3b^-2)/(3b^4a^-4))^3

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