Find the Product (0.5n^5)^2(10n^7)^3

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

Step 1

Apply the product rule to $0.5 n^{5}$ .

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

Step 2

Raise $0.5$ to the power of $2$ .

$0.25 {(n^{5})}^{2} {(10 n^{7})}^{3}$

Step 3

Multiply the exponents in ${(n^{5})}^{2}$ .

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

$0.25 n^{5 \cdot 2} {(10 n^{7})}^{3}$

Multiply $5$ by $2$ .

$0.25 n^{10} {(10 n^{7})}^{3}$

Step 4

Apply the product rule to $10 n^{7}$ .

$0.25 n^{10} (10^{3} {(n^{7})}^{3})$

Step 5

Rewrite using the commutative property of multiplication.

$0.25 \cdot 10^{3} n^{10} {(n^{7})}^{3}$

Step 6

Move the decimal point in $0.25$ to the right by $1$ place and decrease the power of $10^{3}$ by $1$ .

$2.5 \cdot 10^{2} n^{10} {(n^{7})}^{3}$

Step 7

Multiply the exponents in ${(n^{7})}^{3}$ .

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

$2.5 \cdot 10^{2} n^{10} n^{7 \cdot 3}$

Multiply $7$ by $3$ .

$2.5 \cdot 10^{2} n^{10} n^{21}$

Step 8

Multiply $n^{10}$ by $n^{21}$ by adding the exponents.

Move $n^{21}$ .

$2.5 \cdot 10^{2} (n^{21} n^{10})$

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

$2.5 \cdot 10^{2} n^{21 + 10}$

Add $21$ and $10$ .

$2.5 \cdot 10^{2} n^{31}$