Simplify (2ag^2)^4(3a^2g^3)^2

${(2 a g^{2})}^{4} {(3 a^{2} g^{3})}^{2}$

Step 1

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

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

${(2 a)}^{4} {(g^{2})}^{4} {(3 a^{2} g^{3})}^{2}$

Apply the product rule to $2 a$ .

$2^{4} a^{4} {(g^{2})}^{4} {(3 a^{2} g^{3})}^{2}$

Step 2

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

$16 a^{4} {(g^{2})}^{4} {(3 a^{2} g^{3})}^{2}$

Step 3

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

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

$16 a^{4} g^{2 \cdot 4} {(3 a^{2} g^{3})}^{2}$

Multiply $2$ by $4$ .

$16 a^{4} g^{8} {(3 a^{2} g^{3})}^{2}$

Step 4

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

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

$16 a^{4} g^{8} ({(3 a^{2})}^{2} {(g^{3})}^{2})$

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

$16 a^{4} g^{8} (3^{2} {(a^{2})}^{2} {(g^{3})}^{2})$

Step 5

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

$16 a^{4} g^{8} (9 {(a^{2})}^{2} {(g^{3})}^{2})$

Step 6

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

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

$16 a^{4} g^{8} (9 a^{2 \cdot 2} {(g^{3})}^{2})$

Multiply $2$ by $2$ .

$16 a^{4} g^{8} (9 a^{4} {(g^{3})}^{2})$

Step 7

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

Move $a^{4}$ .

$16 (a^{4} a^{4}) g^{8} (9 {(g^{3})}^{2})$

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

$16 a^{4 + 4} g^{8} (9 {(g^{3})}^{2})$

Add $4$ and $4$ .

$16 a^{8} g^{8} (9 {(g^{3})}^{2})$

Step 8

Multiply the exponents in ${(g^{3})}^{2}$ .

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

$16 a^{8} g^{8} (9 g^{3 \cdot 2})$

Multiply $3$ by $2$ .

$16 a^{8} g^{8} (9 g^{6})$

Step 9

Multiply $g^{8}$ by $g^{6}$ by adding the exponents.

Move $g^{6}$ .

$16 a^{8} (g^{6} g^{8}) \cdot 9$

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

$16 a^{8} g^{6 + 8} \cdot 9$

Add $6$ and $8$ .

$16 a^{8} g^{14} \cdot 9$

Step 10

Multiply $9$ by $16$ .

$144 a^{8} g^{14}$

Simplify (2ag^2)^4(3a^2g^3)^2

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