Algebra Examples

$\tan^{2} (\frac{π}{4}) \sec (\frac{π}{4}) + \sec^{3} (\frac{π}{4})$

Step 1

Simplify each term.

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Step 1.1

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$1^{2} \sec (\frac{π}{4}) + \sec^{3} (\frac{π}{4})$

Step 1.2

One to any power is one.

$1 \sec (\frac{π}{4}) + \sec^{3} (\frac{π}{4})$

Step 1.3

Multiply $\sec (\frac{π}{4})$ by $1$ .

$\sec (\frac{π}{4}) + \sec^{3} (\frac{π}{4})$

Step 1.4

The exact value of $\sec (\frac{π}{4})$ is $\frac{2}{\sqrt{2}}$ .

$\frac{2}{\sqrt{2}} + \sec^{3} (\frac{π}{4})$

Step 1.5

Multiply $\frac{2}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} + \sec^{3} (\frac{π}{4})$

Step 1.6

Combine and simplify the denominator.

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Step 1.6.1

Multiply $\frac{2}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}} + \sec^{3} (\frac{π}{4})$

Step 1.6.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{2 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} + \sec^{3} (\frac{π}{4})$

Step 1.6.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{2 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} + \sec^{3} (\frac{π}{4})$

Step 1.6.4

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

$\frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}} + \sec^{3} (\frac{π}{4})$

Step 1.6.5

Add $1$ and $1$ .

$\frac{2 \sqrt{2}}{{\sqrt{2}}^{2}} + \sec^{3} (\frac{π}{4})$

Step 1.6.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 1.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} + \sec^{3} (\frac{π}{4})$

Step 1.6.6.2

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

$\frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}} + \sec^{3} (\frac{π}{4})$

Step 1.6.6.3

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

$\frac{2 \sqrt{2}}{2^{\frac{2}{2}}} + \sec^{3} (\frac{π}{4})$

Step 1.6.6.4

Cancel the common factor of $2$ .

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Step 1.6.6.4.1

Cancel the common factor.

$\frac{2 \sqrt{2}}{2^{\frac{2}{2}}} + \sec^{3} (\frac{π}{4})$

Step 1.6.6.4.2

Rewrite the expression.

$\frac{2 \sqrt{2}}{2^{1}} + \sec^{3} (\frac{π}{4})$

Step 1.6.6.5

Evaluate the exponent.

$\frac{2 \sqrt{2}}{2} + \sec^{3} (\frac{π}{4})$

Step 1.7

Cancel the common factor of $2$ .

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Step 1.7.1

Cancel the common factor.

$\frac{2 \sqrt{2}}{2} + \sec^{3} (\frac{π}{4})$

Step 1.7.2

Divide $\sqrt{2}$ by $1$ .

$\sqrt{2} + \sec^{3} (\frac{π}{4})$

Step 1.8

The exact value of $\sec (\frac{π}{4})$ is $\frac{2}{\sqrt{2}}$ .

$\sqrt{2} + {(\frac{2}{\sqrt{2}})}^{3}$

Step 1.9

Multiply $\frac{2}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sqrt{2} + {(\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})}^{3}$

Step 1.10

Combine and simplify the denominator.

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Step 1.10.1

Multiply $\frac{2}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sqrt{2} + {(\frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}})}^{3}$

Step 1.10.2

Raise $\sqrt{2}$ to the power of $1$ .

$\sqrt{2} + {(\frac{2 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}})}^{3}$

Step 1.10.3

Raise $\sqrt{2}$ to the power of $1$ .

$\sqrt{2} + {(\frac{2 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}})}^{3}$

Step 1.10.4

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

$\sqrt{2} + {(\frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}})}^{3}$

Step 1.10.5

Add $1$ and $1$ .

$\sqrt{2} + {(\frac{2 \sqrt{2}}{{\sqrt{2}}^{2}})}^{3}$

Step 1.10.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 1.10.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\sqrt{2} + {(\frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}})}^{3}$

Step 1.10.6.2

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

$\sqrt{2} + {(\frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}})}^{3}$

Step 1.10.6.3

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

$\sqrt{2} + {(\frac{2 \sqrt{2}}{2^{\frac{2}{2}}})}^{3}$

Step 1.10.6.4

Cancel the common factor of $2$ .

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Step 1.10.6.4.1

Cancel the common factor.

$\sqrt{2} + {(\frac{2 \sqrt{2}}{2^{\frac{2}{2}}})}^{3}$

Step 1.10.6.4.2

Rewrite the expression.

$\sqrt{2} + {(\frac{2 \sqrt{2}}{2^{1}})}^{3}$

Step 1.10.6.5

Evaluate the exponent.

$\sqrt{2} + {(\frac{2 \sqrt{2}}{2})}^{3}$

Step 1.11

Cancel the common factor of $2$ .

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Step 1.11.1

Cancel the common factor.

$\sqrt{2} + {(\frac{2 \sqrt{2}}{2})}^{3}$

Step 1.11.2

Divide $\sqrt{2}$ by $1$ .

$\sqrt{2} + {\sqrt{2}}^{3}$

Step 1.12

Rewrite ${\sqrt{2}}^{3}$ as $\sqrt{2^{3}}$ .

$\sqrt{2} + \sqrt{2^{3}}$

Step 1.13

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

$\sqrt{2} + \sqrt{8}$

Step 1.14

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 1.14.1

Factor $4$ out of $8$ .

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

Step 1.14.2

Rewrite $4$ as $2^{2}$ .

$\sqrt{2} + \sqrt{2^{2} \cdot 2}$

Step 1.15

Pull terms out from under the radical.

$\sqrt{2} + 2 \sqrt{2}$

Step 2

Add $\sqrt{2}$ and $2 \sqrt{2}$ .

$3 \sqrt{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$3 \sqrt{2}$

Decimal Form:

$4.24264068 \dots$