Simplify ((2a^3b^4)^3)/((2ab^2)^5)

$\frac{{(2 a^{3} b^{4})}^{3}}{{(2 a b^{2})}^{5}}$

Step 1

Simplify the numerator.

Apply the product rule to $2 a^{3} b^{4}$ .

$\frac{{(2 a^{3})}^{3} {(b^{4})}^{3}}{{(2 a b^{2})}^{5}}$

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

$\frac{2^{3} {(a^{3})}^{3} {(b^{4})}^{3}}{{(2 a b^{2})}^{5}}$

Raise $2$ to the power of $3$ .

$\frac{8 {(a^{3})}^{3} {(b^{4})}^{3}}{{(2 a b^{2})}^{5}}$

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

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

$\frac{8 a^{3 \cdot 3} {(b^{4})}^{3}}{{(2 a b^{2})}^{5}}$

Multiply $3$ by $3$ .

$\frac{8 a^{9} {(b^{4})}^{3}}{{(2 a b^{2})}^{5}}$

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

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

$\frac{8 a^{9} b^{4 \cdot 3}}{{(2 a b^{2})}^{5}}$

Multiply $4$ by $3$ .

$\frac{8 a^{9} b^{12}}{{(2 a b^{2})}^{5}}$

Step 2

Simplify the denominator.

Apply the product rule to $2 a b^{2}$ .

$\frac{8 a^{9} b^{12}}{{(2 a)}^{5} {(b^{2})}^{5}}$

Apply the product rule to $2 a$ .

$\frac{8 a^{9} b^{12}}{2^{5} a^{5} {(b^{2})}^{5}}$

Raise $2$ to the power of $5$ .

$\frac{8 a^{9} b^{12}}{32 a^{5} {(b^{2})}^{5}}$

Multiply the exponents in ${(b^{2})}^{5}$ .

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

$\frac{8 a^{9} b^{12}}{32 a^{5} b^{2 \cdot 5}}$

Multiply $2$ by $5$ .

$\frac{8 a^{9} b^{12}}{32 a^{5} b^{10}}$

Step 3

Reduce the expression by cancelling the common factors.

Cancel the common factor of $8$ and $32$ .

Factor $8$ out of $8 a^{9} b^{12}$ .

$\frac{8 (a^{9} b^{12})}{32 a^{5} b^{10}}$

Cancel the common factors.

Factor $8$ out of $32 a^{5} b^{10}$ .

$\frac{8 (a^{9} b^{12})}{8 (4 a^{5} b^{10})}$

Cancel the common factor.

$\frac{8 (a^{9} b^{12})}{8 (4 a^{5} b^{10})}$

Rewrite the expression.

$\frac{a^{9} b^{12}}{4 a^{5} b^{10}}$

Cancel the common factor of $a^{9}$ and $a^{5}$ .

Factor $a^{5}$ out of $a^{9} b^{12}$ .

$\frac{a^{5} (a^{4} b^{12})}{4 a^{5} b^{10}}$

Cancel the common factors.

Factor $a^{5}$ out of $4 a^{5} b^{10}$ .

$\frac{a^{5} (a^{4} b^{12})}{a^{5} (4 b^{10})}$

Cancel the common factor.

$\frac{a^{5} (a^{4} b^{12})}{a^{5} (4 b^{10})}$

Rewrite the expression.

$\frac{a^{4} b^{12}}{4 b^{10}}$

Cancel the common factor of $b^{12}$ and $b^{10}$ .

Factor $b^{10}$ out of $a^{4} b^{12}$ .

$\frac{b^{10} (a^{4} b^{2})}{4 b^{10}}$

Cancel the common factors.

Factor $b^{10}$ out of $4 b^{10}$ .

$\frac{b^{10} (a^{4} b^{2})}{b^{10} \cdot 4}$

Cancel the common factor.

$\frac{b^{10} (a^{4} b^{2})}{b^{10} \cdot 4}$

Rewrite the expression.

$\frac{a^{4} b^{2}}{4}$

Simplify ((2a^3b^4)^3)/((2ab^2)^5)

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