Simplify (1/2y^2)^3(2/3y)^2

${(\frac{1}{2} y^{2})}^{3} {(\frac{2}{3} y)}^{2}$

Step 1

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

${(\frac{y^{2}}{2})}^{3} {(\frac{2}{3} y)}^{2}$

Step 2

Simplify the expression.

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

$\frac{{(y^{2})}^{3}}{2^{3}} {(\frac{2}{3} y)}^{2}$

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

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

$\frac{y^{2 \cdot 3}}{2^{3}} {(\frac{2}{3} y)}^{2}$

Multiply $2$ by $3$ .

$\frac{y^{6}}{2^{3}} {(\frac{2}{3} y)}^{2}$

Raise $2$ to the power of $3$ .

$\frac{y^{6}}{8} {(\frac{2}{3} y)}^{2}$

Step 3

Combine $\frac{2}{3}$ and $y$ .

$\frac{y^{6}}{8} {(\frac{2 y}{3})}^{2}$

Step 4

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

Apply the product rule to $\frac{2 y}{3}$ .

$\frac{y^{6}}{8} \cdot \frac{{(2 y)}^{2}}{3^{2}}$

Apply the product rule to $2 y$ .

$\frac{y^{6}}{8} \cdot \frac{2^{2} y^{2}}{3^{2}}$

Step 5

Combine.

$\frac{y^{6} (2^{2} y^{2})}{8 \cdot 3^{2}}$

Step 6

Multiply $y^{6}$ by $y^{2}$ by adding the exponents.

Move $y^{2}$ .

$\frac{y^{2} y^{6} \cdot 2^{2}}{8 \cdot 3^{2}}$

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

$\frac{y^{2 + 6} \cdot 2^{2}}{8 \cdot 3^{2}}$

Add $2$ and $6$ .

$\frac{y^{8} \cdot 2^{2}}{8 \cdot 3^{2}}$

Step 7

Simplify the expression.

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

$\frac{y^{8} \cdot 4}{8 \cdot 3^{2}}$

Raise $3$ to the power of $2$ .

$\frac{y^{8} \cdot 4}{8 \cdot 9}$

Multiply $8$ by $9$ .

$\frac{y^{8} \cdot 4}{72}$

Step 8

Cancel the common factor of $4$ and $72$ .

Factor $4$ out of $y^{8} \cdot 4$ .

$\frac{4 \cdot y^{8}}{72}$

Cancel the common factors.

Factor $4$ out of $72$ .

$\frac{4 \cdot y^{8}}{4 \cdot 18}$

Cancel the common factor.

$\frac{4 \cdot y^{8}}{4 \cdot 18}$

Rewrite the expression.

$\frac{y^{8}}{18}$

Simplify (1/2y^2)^3(2/3y)^2

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