Simplify (1/2j^2)^5(2/5j^3)^2

${(\frac{1}{2} j^{2})}^{5} {(\frac{2}{5} j^{3})}^{2}$

Step 1

Combine $\frac{1}{2}$ and $j^{2}$ .

${(\frac{j^{2}}{2})}^{5} {(\frac{2}{5} j^{3})}^{2}$

Step 2

Simplify the expression.

Apply the product rule to $\frac{j^{2}}{2}$ .

$\frac{{(j^{2})}^{5}}{2^{5}} {(\frac{2}{5} j^{3})}^{2}$

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

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

$\frac{j^{2 \cdot 5}}{2^{5}} {(\frac{2}{5} j^{3})}^{2}$

Multiply $2$ by $5$ .

$\frac{j^{10}}{2^{5}} {(\frac{2}{5} j^{3})}^{2}$

Raise $2$ to the power of $5$ .

$\frac{j^{10}}{32} {(\frac{2}{5} j^{3})}^{2}$

Step 3

Combine $\frac{2}{5}$ and $j^{3}$ .

$\frac{j^{10}}{32} {(\frac{2 j^{3}}{5})}^{2}$

Step 4

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

Apply the product rule to $\frac{2 j^{3}}{5}$ .

$\frac{j^{10}}{32} \cdot \frac{{(2 j^{3})}^{2}}{5^{2}}$

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

$\frac{j^{10}}{32} \cdot \frac{2^{2} {(j^{3})}^{2}}{5^{2}}$

Step 5

Combine.

$\frac{j^{10} (2^{2} {(j^{3})}^{2})}{32 \cdot 5^{2}}$

Step 6

Simplify the numerator.

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

$\frac{j^{10} \cdot 4 {(j^{3})}^{2}}{32 \cdot 5^{2}}$

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

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

$\frac{j^{10} \cdot 4 j^{3 \cdot 2}}{32 \cdot 5^{2}}$

Multiply $3$ by $2$ .

$\frac{j^{10} \cdot 4 j^{6}}{32 \cdot 5^{2}}$

Multiply $j^{10}$ by $j^{6}$ by adding the exponents.

Move $j^{6}$ .

$\frac{j^{6} j^{10} \cdot 4}{32 \cdot 5^{2}}$

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

$\frac{j^{6 + 10} \cdot 4}{32 \cdot 5^{2}}$

Add $6$ and $10$ .

$\frac{j^{16} \cdot 4}{32 \cdot 5^{2}}$

Step 7

Reduce the expression by cancelling the common factors.

Raise $5$ to the power of $2$ .

$\frac{j^{16} \cdot 4}{32 \cdot 25}$

Multiply $32$ by $25$ .

$\frac{j^{16} \cdot 4}{800}$

Cancel the common factor of $4$ and $800$ .

Factor $4$ out of $j^{16} \cdot 4$ .

$\frac{4 \cdot j^{16}}{800}$

Cancel the common factors.

Factor $4$ out of $800$ .

$\frac{4 \cdot j^{16}}{4 \cdot 200}$

Cancel the common factor.

$\frac{4 \cdot j^{16}}{4 \cdot 200}$

Rewrite the expression.

$\frac{j^{16}}{200}$

Simplify (1/2j^2)^5(2/5j^3)^2

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