Calculus Examples

${(2 t^{2} - 1)}^{2}$

Step 1

Rewrite ${(2 t^{2} - 1)}^{2}$ as $(2 t^{2} - 1) (2 t^{2} - 1)$ .

$(2 t^{2} - 1) (2 t^{2} - 1)$

Step 2

Expand $(2 t^{2} - 1) (2 t^{2} - 1)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$2 t^{2} (2 t^{2} - 1) - 1 (2 t^{2} - 1)$

Step 2.2

Apply the distributive property.

$2 t^{2} (2 t^{2}) + 2 t^{2} \cdot - 1 - 1 (2 t^{2} - 1)$

Step 2.3

Apply the distributive property.

$2 t^{2} (2 t^{2}) + 2 t^{2} \cdot - 1 - 1 (2 t^{2}) - 1 \cdot - 1$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite using the commutative property of multiplication.

$2 \cdot 2 t^{2} t^{2} + 2 t^{2} \cdot - 1 - 1 (2 t^{2}) - 1 \cdot - 1$

Step 3.1.2

Multiply $t^{2}$ by $t^{2}$ by adding the exponents.

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Step 3.1.2.1

Move $t^{2}$ .

$2 \cdot 2 (t^{2} t^{2}) + 2 t^{2} \cdot - 1 - 1 (2 t^{2}) - 1 \cdot - 1$

Step 3.1.2.2

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

$2 \cdot 2 t^{2 + 2} + 2 t^{2} \cdot - 1 - 1 (2 t^{2}) - 1 \cdot - 1$

Step 3.1.2.3

Add $2$ and $2$ .

$2 \cdot 2 t^{4} + 2 t^{2} \cdot - 1 - 1 (2 t^{2}) - 1 \cdot - 1$

Step 3.1.3

Multiply $2$ by $2$ .

$4 t^{4} + 2 t^{2} \cdot - 1 - 1 (2 t^{2}) - 1 \cdot - 1$

Step 3.1.4

Multiply $- 1$ by $2$ .

$4 t^{4} - 2 t^{2} - 1 (2 t^{2}) - 1 \cdot - 1$

Step 3.1.5

Multiply $2$ by $- 1$ .

$4 t^{4} - 2 t^{2} - 2 t^{2} - 1 \cdot - 1$

Step 3.1.6

Multiply $- 1$ by $- 1$ .

$4 t^{4} - 2 t^{2} - 2 t^{2} + 1$

Step 3.2

Subtract $2 t^{2}$ from $- 2 t^{2}$ .

$4 t^{4} - 4 t^{2} + 1$