Precalculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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

Multiply $1$ by $1$ .

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

Step 3.1.2

Multiply $- t^{2}$ by $1$ .

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

Step 3.1.3

Multiply $- 1$ by $1$ .

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

Step 3.1.4

Rewrite using the commutative property of multiplication.

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

Step 3.1.5

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

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Step 3.1.5.1

Move $t^{2}$ .

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

Step 3.1.5.2

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

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

Step 3.1.5.3

Add $2$ and $2$ .

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

Step 3.1.6

Multiply $- 1$ by $- 1$ .

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

Step 3.1.7

Multiply $t^{4}$ by $1$ .

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

Step 3.2

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

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

Step 4

Simplify the expression.

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Step 4.1

Apply the product rule to $2 t$ .

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

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

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

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

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $1$ by $1$ .

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

Step 6.1.2

Multiply $t^{2}$ by $1$ .

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

Step 6.1.3

Multiply $t^{2}$ by $1$ .

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

Step 6.1.4

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

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Step 6.1.4.1

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

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

Step 6.1.4.2

Add $2$ and $2$ .

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

Step 6.2

Add $t^{2}$ and $t^{2}$ .

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

Step 7

Add $1$ and $1$ .

$\sqrt{- 2 t^{2} + t^{4} + 4 t^{2} + 2 + 2 t^{2} + t^{4}}$

Step 8

Add $- 2 t^{2}$ and $4 t^{2}$ .

$\sqrt{t^{4} + 2 t^{2} + 2 + 2 t^{2} + t^{4}}$

Step 9

Add $t^{4}$ and $t^{4}$ .

$\sqrt{2 t^{4} + 2 t^{2} + 2 + 2 t^{2}}$

Step 10

Add $2 t^{2}$ and $2 t^{2}$ .

$\sqrt{2 t^{4} + 4 t^{2} + 2}$

Step 11

Rewrite $2 t^{4} + 4 t^{2} + 2$ in a factored form.

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Step 11.1

Factor $2$ out of $2 t^{4} + 4 t^{2} + 2$ .

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Step 11.1.1

Factor $2$ out of $2 t^{4}$ .

$\sqrt{2 (t^{4}) + 4 t^{2} + 2}$

Step 11.1.2

Factor $2$ out of $4 t^{2}$ .

$\sqrt{2 (t^{4}) + 2 (2 t^{2}) + 2}$

Step 11.1.3

Factor $2$ out of $2$ .

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

Step 11.1.4

Factor $2$ out of $2 (t^{4}) + 2 (2 t^{2})$ .

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

Step 11.1.5

Factor $2$ out of $2 (t^{4} + 2 t^{2}) + 2 (1)$ .

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

Step 11.2

Rewrite $t^{4}$ as ${(t^{2})}^{2}$ .

$\sqrt{2 ({(t^{2})}^{2} + 2 t^{2} + 1)}$

Step 11.3

Let $u = t^{2}$ . Substitute $u$ for all occurrences of $t^{2}$ .

$\sqrt{2 (u^{2} + 2 u + 1)}$

Step 11.4

Factor using the perfect square rule.

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Step 11.4.1

Rewrite $1$ as $1^{2}$ .

$\sqrt{2 (u^{2} + 2 u + 1^{2})}$

Step 11.4.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 u = 2 \cdot u \cdot 1$

Step 11.4.3

Rewrite the polynomial.

$\sqrt{2 (u^{2} + 2 \cdot u \cdot 1 + 1^{2})}$

Step 11.4.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = u$ and $b = 1$ .

$\sqrt{2 {(u + 1)}^{2}}$

Step 11.5

Replace all occurrences of $u$ with $t^{2}$ .

$\sqrt{2 {(t^{2} + 1)}^{2}}$

Step 12

Rewrite $2 {(t^{2} + 1)}^{2}$ as ${(t^{2} + 1^{2})}^{2} \cdot 2$ .

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Step 12.1

Reorder $2$ and ${(t^{2} + 1)}^{2}$ .

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

Step 12.2

Rewrite $1$ as $1^{2}$ .

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

Step 13

Pull terms out from under the radical.

$(t^{2} + 1^{2}) \sqrt{2}$

Step 14

One to any power is one.

$(t^{2} + 1) \sqrt{2}$