Solve (0.5n^5)^2(10n^7)^3

$$( 0.5 n ^ { 5 } ) ^ { 2 } ( 10 n ^ { 7 } ) ^ { 3 }$$

Evaluate

$$250n^{31}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(0.5n^{5}\right)^{2}\times \left(10n^{7}\right)^{3}$$

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

$$0.5^{2}\left(n^{5}\right)^{2}\times 10^{3}\left(n^{7}\right)^{3}$$

Use the Commutative Property of Multiplication.

$$0.5^{2}\times 10^{3}\left(n^{5}\right)^{2}\left(n^{7}\right)^{3}$$

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

$$0.5^{2}\times 10^{3}n^{5\times 2}n^{7\times 3}$$

Multiply 5 times 2.

$$0.5^{2}\times 10^{3}n^{10}n^{7\times 3}$$

Multiply 7 times 3.

$$0.5^{2}\times 10^{3}n^{10}n^{21}$$

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

$$0.5^{2}\times 10^{3}n^{10+21}$$

Add the exponents 10 and 21.

$$0.5^{2}\times 10^{3}n^{31}$$

Raise 0.5 to the power 2.

$$0.25\times 10^{3}n^{31}$$

Raise 10 to the power 3.

$$0.25\times 1000n^{31}$$

Multiply 0.25 times 1000.

$$250n^{31}$$

Expand

$$250n^{31}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(0.5n^{5}\right)^{2}\times \left(10n^{7}\right)^{3}$$

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

$$0.5^{2}\left(n^{5}\right)^{2}\times 10^{3}\left(n^{7}\right)^{3}$$

Use the Commutative Property of Multiplication.

$$0.5^{2}\times 10^{3}\left(n^{5}\right)^{2}\left(n^{7}\right)^{3}$$

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

$$0.5^{2}\times 10^{3}n^{5\times 2}n^{7\times 3}$$

Multiply 5 times 2.

$$0.5^{2}\times 10^{3}n^{10}n^{7\times 3}$$

Multiply 7 times 3.

$$0.5^{2}\times 10^{3}n^{10}n^{21}$$

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

$$0.5^{2}\times 10^{3}n^{10+21}$$

Add the exponents 10 and 21.

$$0.5^{2}\times 10^{3}n^{31}$$

Raise 0.5 to the power 2.

$$0.25\times 10^{3}n^{31}$$

Raise 10 to the power 3.

$$0.25\times 1000n^{31}$$

Multiply 0.25 times 1000.

$$250n^{31}$$