Simplify (2a^5b^5)^3(3a^5b^3)^3

${(2 a^{5} b^{5})}^{3} {(3 a^{5} b^{3})}^{3}$

Step 1

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

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

${(2 a^{5})}^{3} {(b^{5})}^{3} {(3 a^{5} b^{3})}^{3}$

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

$2^{3} {(a^{5})}^{3} {(b^{5})}^{3} {(3 a^{5} b^{3})}^{3}$

Step 2

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

$8 {(a^{5})}^{3} {(b^{5})}^{3} {(3 a^{5} b^{3})}^{3}$

Step 3

Multiply the exponents in ${(a^{5})}^{3}$ .

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

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

Multiply $5$ by $3$ .

$8 a^{15} {(b^{5})}^{3} {(3 a^{5} b^{3})}^{3}$

Step 4

Multiply the exponents in ${(b^{5})}^{3}$ .

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

$8 a^{15} b^{5 \cdot 3} {(3 a^{5} b^{3})}^{3}$

Multiply $5$ by $3$ .

$8 a^{15} b^{15} {(3 a^{5} b^{3})}^{3}$

Step 5

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

Apply the product rule to $3 a^{5} b^{3}$ .

$8 a^{15} b^{15} ({(3 a^{5})}^{3} {(b^{3})}^{3})$

Apply the product rule to $3 a^{5}$ .

$8 a^{15} b^{15} (3^{3} {(a^{5})}^{3} {(b^{3})}^{3})$

Step 6

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

$8 a^{15} b^{15} (27 {(a^{5})}^{3} {(b^{3})}^{3})$

Step 7

Multiply the exponents in ${(a^{5})}^{3}$ .

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

$8 a^{15} b^{15} (27 a^{5 \cdot 3} {(b^{3})}^{3})$

Multiply $5$ by $3$ .

$8 a^{15} b^{15} (27 a^{15} {(b^{3})}^{3})$

Step 8

Multiply $a^{15}$ by $a^{15}$ by adding the exponents.

Move $a^{15}$ .

$8 (a^{15} a^{15}) b^{15} (27 {(b^{3})}^{3})$

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

$8 a^{15 + 15} b^{15} (27 {(b^{3})}^{3})$

Add $15$ and $15$ .

$8 a^{30} b^{15} (27 {(b^{3})}^{3})$

Step 9

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

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

$8 a^{30} b^{15} (27 b^{3 \cdot 3})$

Multiply $3$ by $3$ .

$8 a^{30} b^{15} (27 b^{9})$

Step 10

Multiply $b^{15}$ by $b^{9}$ by adding the exponents.

Move $b^{9}$ .

$8 a^{30} (b^{9} b^{15}) \cdot 27$

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

$8 a^{30} b^{9 + 15} \cdot 27$

Add $9$ and $15$ .

$8 a^{30} b^{24} \cdot 27$

Step 11

Multiply $27$ by $8$ .

$216 a^{30} b^{24}$

Simplify (2a^5b^5)^3(3a^5b^3)^3

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