Trigonometry Examples

$\sqrt{p^{2} + 1} = 5$

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{p^{2} + 1}}^{2} = 5^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

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

${({(p^{2} + 1)}^{\frac{1}{2}})}^{2} = 5^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(p^{2} + 1)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(p^{2} + 1)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(p^{2} + 1)}^{\frac{1}{2} \cdot 2} = 5^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(p^{2} + 1)}^{\frac{1}{2} \cdot 2} = 5^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(p^{2} + 1)}^{1} = 5^{2}$

Step 2.2.1.2

Simplify.

$p^{2} + 1 = 5^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Raise $5$ to the power of $2$ .

$p^{2} + 1 = 25$

Step 3

Solve for $p$ .

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Step 3.1

Move all terms not containing $p$ to the right side of the equation.

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Step 3.1.1

Subtract $1$ from both sides of the equation.

$p^{2} = 25 - 1$

Step 3.1.2

Subtract $1$ from $25$ .

$p^{2} = 24$

Step 3.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$p = \pm \sqrt{24}$

Step 3.3

Simplify $\pm \sqrt{24}$ .

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Step 3.3.1

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 3.3.1.1

Factor $4$ out of $24$ .

$p = \pm \sqrt{4 (6)}$

Step 3.3.1.2

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

$p = \pm \sqrt{2^{2} \cdot 6}$

Step 3.3.2

Pull terms out from under the radical.

$p = \pm 2 \sqrt{6}$

Step 3.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.4.1

First, use the positive value of the $\pm$ to find the first solution.

$p = 2 \sqrt{6}$

Step 3.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$p = - 2 \sqrt{6}$

Step 3.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$p = 2 \sqrt{6}, - 2 \sqrt{6}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$p = 2 \sqrt{6}, - 2 \sqrt{6}$

Decimal Form:

$p = 4.89897948 \dots, - 4.89897948 \dots$