Solve (a^3b)^4(ab^6)^2

$$( a ^ { 3 } b ) ^ { 4 } ( a b ^ { 6 } ) ^ { 2 }$$

Evaluate

$$a^{14}b^{16}$$

Steps Using Product of Powers Property

Use the rules of exponents to simplify the expression.

$$\left(a^{3}b^{1}\right)^{4}\left(a^{1}b^{6}\right)^{2}$$

To raise the product of two or more numbers to a power, raise each number to the power and take their product.

$$\left(a^{3}\right)^{4}\left(b^{1}\right)^{4}\left(a^{1}\right)^{2}\left(b^{6}\right)^{2}$$

Use the Commutative Property of Multiplication.

$$\left(a^{3}\right)^{4}\left(a^{1}\right)^{2}\left(b^{1}\right)^{4}\left(b^{6}\right)^{2}$$

To raise a power to another power, multiply the exponents.

$$a^{3\times 4}a^{2}b^{4}b^{6\times 2}$$

Multiply 3 times 4.

$$a^{12}a^{2}b^{4}b^{6\times 2}$$

Multiply 6 times 2.

$$a^{12}a^{2}b^{4}b^{12}$$

To multiply powers of the same base, add their exponents.

$$a^{12+2}b^{4+12}$$

Add the exponents 12 and 2.

$$a^{14}b^{4+12}$$

Add the exponents 4 and 12.

$$a^{14}b^{16}$$

Expand

$$a^{14}b^{16}$$

Steps Using Product of Powers Property

Use the rules of exponents to simplify the expression.

$$\left(a^{3}b^{1}\right)^{4}\left(a^{1}b^{6}\right)^{2}$$

To raise the product of two or more numbers to a power, raise each number to the power and take their product.

$$\left(a^{3}\right)^{4}\left(b^{1}\right)^{4}\left(a^{1}\right)^{2}\left(b^{6}\right)^{2}$$

Use the Commutative Property of Multiplication.

$$\left(a^{3}\right)^{4}\left(a^{1}\right)^{2}\left(b^{1}\right)^{4}\left(b^{6}\right)^{2}$$

To raise a power to another power, multiply the exponents.

$$a^{3\times 4}a^{2}b^{4}b^{6\times 2}$$

Multiply 3 times 4.

$$a^{12}a^{2}b^{4}b^{6\times 2}$$

Multiply 6 times 2.

$$a^{12}a^{2}b^{4}b^{12}$$

To multiply powers of the same base, add their exponents.

$$a^{12+2}b^{4+12}$$

Add the exponents 12 and 2.

$$a^{14}b^{4+12}$$

Add the exponents 4 and 12.

$$a^{14}b^{16}$$