Solve (ac^{2})^5*(a^3c^2)^4

$$( a c ^ { 2 } ) ^ { 5 } \times ( a ^ { 3 } c ^ { 2 } ) ^ { 4 }$$

Evaluate

$$a^{17}c^{18}$$

Steps Using Product of Powers Property

Use the rules of exponents to simplify the expression.

$$\left(a^{1}c^{2}\right)^{5}\left(a^{3}c^{2}\right)^{4}$$

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

$$\left(a^{1}\right)^{5}\left(c^{2}\right)^{5}\left(a^{3}\right)^{4}\left(c^{2}\right)^{4}$$

Use the Commutative Property of Multiplication.

$$\left(a^{1}\right)^{5}\left(a^{3}\right)^{4}\left(c^{2}\right)^{5}\left(c^{2}\right)^{4}$$

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

$$a^{5}a^{3\times 4}c^{2\times 5}c^{2\times 4}$$

Multiply 3 times 4.

$$a^{5}a^{12}c^{2\times 5}c^{2\times 4}$$

Multiply 2 times 5.

$$a^{5}a^{12}c^{10}c^{2\times 4}$$

Multiply 2 times 4.

$$a^{5}a^{12}c^{10}c^{8}$$

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

$$a^{5+12}c^{10+8}$$

Add the exponents 5 and 12.

$$a^{17}c^{10+8}$$

Add the exponents 10 and 8.

$$a^{17}c^{18}$$

Expand

$$a^{17}c^{18}$$

Steps Using Product of Powers Property

Use the rules of exponents to simplify the expression.

$$\left(a^{1}c^{2}\right)^{5}\left(a^{3}c^{2}\right)^{4}$$

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

$$\left(a^{1}\right)^{5}\left(c^{2}\right)^{5}\left(a^{3}\right)^{4}\left(c^{2}\right)^{4}$$

Use the Commutative Property of Multiplication.

$$\left(a^{1}\right)^{5}\left(a^{3}\right)^{4}\left(c^{2}\right)^{5}\left(c^{2}\right)^{4}$$

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

$$a^{5}a^{3\times 4}c^{2\times 5}c^{2\times 4}$$

Multiply 3 times 4.

$$a^{5}a^{12}c^{2\times 5}c^{2\times 4}$$

Multiply 2 times 5.

$$a^{5}a^{12}c^{10}c^{2\times 4}$$

Multiply 2 times 4.

$$a^{5}a^{12}c^{10}c^{8}$$

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

$$a^{5+12}c^{10+8}$$

Add the exponents 5 and 12.

$$a^{17}c^{10+8}$$

Add the exponents 10 and 8.

$$a^{17}c^{18}$$