Simplify ((y^3)/4-1/(y^3))^2

${(\frac{y^{3}}{4} - \frac{1}{y^{3}})}^{2}$

Step 1

Rewrite ${(\frac{y^{3}}{4} - \frac{1}{y^{3}})}^{2}$ as $(\frac{y^{3}}{4} - \frac{1}{y^{3}}) (\frac{y^{3}}{4} - \frac{1}{y^{3}})$ .

$(\frac{y^{3}}{4} - \frac{1}{y^{3}}) (\frac{y^{3}}{4} - \frac{1}{y^{3}})$

Step 2

Expand $(\frac{y^{3}}{4} - \frac{1}{y^{3}}) (\frac{y^{3}}{4} - \frac{1}{y^{3}})$ using the FOIL Method.

Apply the distributive property.

$\frac{y^{3}}{4} (\frac{y^{3}}{4} - \frac{1}{y^{3}}) - \frac{1}{y^{3}} (\frac{y^{3}}{4} - \frac{1}{y^{3}})$

Apply the distributive property.

$\frac{y^{3}}{4} \cdot \frac{y^{3}}{4} + \frac{y^{3}}{4} (- \frac{1}{y^{3}}) - \frac{1}{y^{3}} (\frac{y^{3}}{4} - \frac{1}{y^{3}})$

Apply the distributive property.

$\frac{y^{3}}{4} \cdot \frac{y^{3}}{4} + \frac{y^{3}}{4} (- \frac{1}{y^{3}}) - \frac{1}{y^{3}} \cdot \frac{y^{3}}{4} - \frac{1}{y^{3}} (- \frac{1}{y^{3}})$

Step 3

Simplify and combine like terms.

Simplify each term.

Combine.

$\frac{y^{3} y^{3}}{4 \cdot 4} + \frac{y^{3}}{4} (- \frac{1}{y^{3}}) - \frac{1}{y^{3}} \cdot \frac{y^{3}}{4} - \frac{1}{y^{3}} (- \frac{1}{y^{3}})$

Multiply $y^{3}$ by $y^{3}$ by adding the exponents.

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

$\frac{y^{3 + 3}}{4 \cdot 4} + \frac{y^{3}}{4} (- \frac{1}{y^{3}}) - \frac{1}{y^{3}} \cdot \frac{y^{3}}{4} - \frac{1}{y^{3}} (- \frac{1}{y^{3}})$

Add $3$ and $3$ .

$\frac{y^{6}}{4 \cdot 4} + \frac{y^{3}}{4} (- \frac{1}{y^{3}}) - \frac{1}{y^{3}} \cdot \frac{y^{3}}{4} - \frac{1}{y^{3}} (- \frac{1}{y^{3}})$

Multiply $4$ by $4$ .

$\frac{y^{6}}{16} + \frac{y^{3}}{4} (- \frac{1}{y^{3}}) - \frac{1}{y^{3}} \cdot \frac{y^{3}}{4} - \frac{1}{y^{3}} (- \frac{1}{y^{3}})$

Rewrite using the commutative property of multiplication.

$\frac{y^{6}}{16} - \frac{y^{3}}{4} \cdot \frac{1}{y^{3}} - \frac{1}{y^{3}} \cdot \frac{y^{3}}{4} - \frac{1}{y^{3}} (- \frac{1}{y^{3}})$

Cancel the common factor of $y^{3}$ .

Move the leading negative in $- \frac{y^{3}}{4}$ into the numerator.

$\frac{y^{6}}{16} + \frac{- y^{3}}{4} \cdot \frac{1}{y^{3}} - \frac{1}{y^{3}} \cdot \frac{y^{3}}{4} - \frac{1}{y^{3}} (- \frac{1}{y^{3}})$

Factor $y^{3}$ out of $- y^{3}$ .

$\frac{y^{6}}{16} + \frac{y^{3} \cdot - 1}{4} \cdot \frac{1}{y^{3}} - \frac{1}{y^{3}} \cdot \frac{y^{3}}{4} - \frac{1}{y^{3}} (- \frac{1}{y^{3}})$

Cancel the common factor.

$\frac{y^{6}}{16} + \frac{y^{3} \cdot - 1}{4} \cdot \frac{1}{y^{3}} - \frac{1}{y^{3}} \cdot \frac{y^{3}}{4} - \frac{1}{y^{3}} (- \frac{1}{y^{3}})$

Rewrite the expression.

$\frac{y^{6}}{16} + \frac{- 1}{4} - \frac{1}{y^{3}} \cdot \frac{y^{3}}{4} - \frac{1}{y^{3}} (- \frac{1}{y^{3}})$

Move the negative in front of the fraction.

$\frac{y^{6}}{16} - \frac{1}{4} - \frac{1}{y^{3}} \cdot \frac{y^{3}}{4} - \frac{1}{y^{3}} (- \frac{1}{y^{3}})$

Cancel the common factor of $y^{3}$ .

Move the leading negative in $- \frac{1}{y^{3}}$ into the numerator.

$\frac{y^{6}}{16} - \frac{1}{4} + \frac{- 1}{y^{3}} \cdot \frac{y^{3}}{4} - \frac{1}{y^{3}} (- \frac{1}{y^{3}})$

Cancel the common factor.

$\frac{y^{6}}{16} - \frac{1}{4} + \frac{- 1}{y^{3}} \cdot \frac{y^{3}}{4} - \frac{1}{y^{3}} (- \frac{1}{y^{3}})$

Rewrite the expression.

$\frac{y^{6}}{16} - \frac{1}{4} - 1 (\frac{1}{4}) - \frac{1}{y^{3}} (- \frac{1}{y^{3}})$

Rewrite $- 1 (\frac{1}{4})$ as $- (\frac{1}{4})$ .

$\frac{y^{6}}{16} - \frac{1}{4} - (\frac{1}{4}) - \frac{1}{y^{3}} (- \frac{1}{y^{3}})$

Multiply $- \frac{1}{y^{3}} (- \frac{1}{y^{3}})$ .

Multiply $- 1$ by $- 1$ .

$\frac{y^{6}}{16} - \frac{1}{4} - \frac{1}{4} + 1 \frac{1}{y^{3}} \frac{1}{y^{3}}$

Multiply $\frac{1}{y^{3}}$ by $1$ .

$\frac{y^{6}}{16} - \frac{1}{4} - \frac{1}{4} + \frac{1}{y^{3}} \cdot \frac{1}{y^{3}}$

Multiply $\frac{1}{y^{3}}$ by $\frac{1}{y^{3}}$ .

$\frac{y^{6}}{16} - \frac{1}{4} - \frac{1}{4} + \frac{1}{y^{3} y^{3}}$

Multiply $y^{3}$ by $y^{3}$ by adding the exponents.

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

$\frac{y^{6}}{16} - \frac{1}{4} - \frac{1}{4} + \frac{1}{y^{3 + 3}}$

Add $3$ and $3$ .

$\frac{y^{6}}{16} - \frac{1}{4} - \frac{1}{4} + \frac{1}{y^{6}}$

Subtract $\frac{1}{4}$ from $- \frac{1}{4}$ .

$\frac{y^{6}}{16} - 2 (\frac{1}{4}) + \frac{1}{y^{6}}$

Step 4

Simplify each term.

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

$\frac{y^{6}}{16} + 2 (- 1) \frac{1}{4} + \frac{1}{y^{6}}$

Factor $2$ out of $4$ .

$\frac{y^{6}}{16} + 2 \cdot - 1 \frac{1}{2 \cdot 2} + \frac{1}{y^{6}}$

Cancel the common factor.

$\frac{y^{6}}{16} + 2 \cdot - 1 \frac{1}{2 \cdot 2} + \frac{1}{y^{6}}$

Rewrite the expression.

$\frac{y^{6}}{16} - 1 (\frac{1}{2}) + \frac{1}{y^{6}}$

Rewrite $- 1 (\frac{1}{2})$ as $- (\frac{1}{2})$ .

$\frac{y^{6}}{16} - \frac{1}{2} + \frac{1}{y^{6}}$

Simplify ((y^3)/4-1/(y^3))^2

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