Basic Math Examples

$p - 4^{4} \sqrt{p} - 5 = 0$

Step 1

Move all terms not containing $- 4^{4} \sqrt{p}$ to the right side of the equation.

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Step 1.1

Subtract $p$ from both sides of the equation.

$- 4^{4} \sqrt{p} - 5 = - p$

Step 1.2

Add $5$ to both sides of the equation.

$- 4^{4} \sqrt{p} = - p + 5$

Step 2

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

${(- 4^{4} \sqrt{p})}^{2} = {(- p + 5)}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

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

${(- 4^{4} p^{\frac{1}{2}})}^{2} = {(- p + 5)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${(- 4^{4} p^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Raise $4$ to the power of $4$ .

${(- 1 \cdot 256 p^{\frac{1}{2}})}^{2} = {(- p + 5)}^{2}$

Step 3.2.1.2

Multiply $- 1$ by $256$ .

${(- 256 p^{\frac{1}{2}})}^{2} = {(- p + 5)}^{2}$

Step 3.2.1.3

Apply the product rule to $- 256 p^{\frac{1}{2}}$ .

${(- 256)}^{2} {(p^{\frac{1}{2}})}^{2} = {(- p + 5)}^{2}$

Step 3.2.1.4

Raise $- 256$ to the power of $2$ .

$65536 {(p^{\frac{1}{2}})}^{2} = {(- p + 5)}^{2}$

Step 3.2.1.5

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

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Step 3.2.1.5.1

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

$65536 p^{\frac{1}{2} \cdot 2} = {(- p + 5)}^{2}$

Step 3.2.1.5.2

Cancel the common factor of $2$ .

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Step 3.2.1.5.2.1

Cancel the common factor.

$65536 p^{\frac{1}{2} \cdot 2} = {(- p + 5)}^{2}$

Step 3.2.1.5.2.2

Rewrite the expression.

$65536 p^{1} = {(- p + 5)}^{2}$

Step 3.2.1.6

Simplify.

$65536 p = {(- p + 5)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(- p + 5)}^{2}$ .

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Step 3.3.1.1

Rewrite ${(- p + 5)}^{2}$ as $(- p + 5) (- p + 5)$ .

$65536 p = (- p + 5) (- p + 5)$

Step 3.3.1.2

Expand $(- p + 5) (- p + 5)$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$65536 p = - p (- p + 5) + 5 (- p + 5)$

Step 3.3.1.2.2

Apply the distributive property.

$65536 p = - p (- p) - p \cdot 5 + 5 (- p + 5)$

Step 3.3.1.2.3

Apply the distributive property.

$65536 p = - p (- p) - p \cdot 5 + 5 (- p) + 5 \cdot 5$

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$65536 p = - 1 \cdot - 1 p \cdot p - p \cdot 5 + 5 (- p) + 5 \cdot 5$

Step 3.3.1.3.1.2

Multiply $p$ by $p$ by adding the exponents.

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Step 3.3.1.3.1.2.1

Move $p$ .

$65536 p = - 1 \cdot - 1 (p \cdot p) - p \cdot 5 + 5 (- p) + 5 \cdot 5$

Step 3.3.1.3.1.2.2

Multiply $p$ by $p$ .

$65536 p = - 1 \cdot - 1 p^{2} - p \cdot 5 + 5 (- p) + 5 \cdot 5$

Step 3.3.1.3.1.3

Multiply $- 1$ by $- 1$ .

$65536 p = 1 p^{2} - p \cdot 5 + 5 (- p) + 5 \cdot 5$

Step 3.3.1.3.1.4

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

$65536 p = p^{2} - p \cdot 5 + 5 (- p) + 5 \cdot 5$

Step 3.3.1.3.1.5

Multiply $5$ by $- 1$ .

$65536 p = p^{2} - 5 p + 5 (- p) + 5 \cdot 5$

Step 3.3.1.3.1.6

Multiply $- 1$ by $5$ .

$65536 p = p^{2} - 5 p - 5 p + 5 \cdot 5$

Step 3.3.1.3.1.7

Multiply $5$ by $5$ .

$65536 p = p^{2} - 5 p - 5 p + 25$

Step 3.3.1.3.2

Subtract $5 p$ from $- 5 p$ .

$65536 p = p^{2} - 10 p + 25$

Step 4

Solve for $p$ .

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Step 4.1

Since $p$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$p^{2} - 10 p + 25 = 65536 p$

Step 4.2

Move all terms containing $p$ to the left side of the equation.

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Step 4.2.1

Subtract $65536 p$ from both sides of the equation.

$p^{2} - 10 p + 25 - 65536 p = 0$

Step 4.2.2

Subtract $65536 p$ from $- 10 p$ .

$p^{2} - 65546 p + 25 = 0$

Step 4.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.4

Substitute the values $a = 1$ , $b = - 65546$ , and $c = 25$ into the quadratic formula and solve for $p$ .

$\frac{65546 \pm \sqrt{{(- 65546)}^{2} - 4 \cdot (1 \cdot 25)}}{2 \cdot 1}$

Step 4.5

Simplify.

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Step 4.5.1

Simplify the numerator.

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Step 4.5.1.1

Raise $- 65546$ to the power of $2$ .

$p = \frac{65546 \pm \sqrt{4296278116 - 4 \cdot 1 \cdot 25}}{2 \cdot 1}$

Step 4.5.1.2

Multiply $- 4 \cdot 1 \cdot 25$ .

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Step 4.5.1.2.1

Multiply $- 4$ by $1$ .

$p = \frac{65546 \pm \sqrt{4296278116 - 4 \cdot 25}}{2 \cdot 1}$

Step 4.5.1.2.2

Multiply $- 4$ by $25$ .

$p = \frac{65546 \pm \sqrt{4296278116 - 100}}{2 \cdot 1}$

Step 4.5.1.3

Subtract $100$ from $4296278116$ .

$p = \frac{65546 \pm \sqrt{4296278016}}{2 \cdot 1}$

Step 4.5.1.4

Rewrite $4296278016$ as $1536^{2} \cdot 1821$ .

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Step 4.5.1.4.1

Factor $2359296$ out of $4296278016$ .

$p = \frac{65546 \pm \sqrt{2359296 (1821)}}{2 \cdot 1}$

Step 4.5.1.4.2

Rewrite $2359296$ as $1536^{2}$ .

$p = \frac{65546 \pm \sqrt{1536^{2} \cdot 1821}}{2 \cdot 1}$

Step 4.5.1.5

Pull terms out from under the radical.

$p = \frac{65546 \pm 1536 \sqrt{1821}}{2 \cdot 1}$

Step 4.5.2

Multiply $2$ by $1$ .

$p = \frac{65546 \pm 1536 \sqrt{1821}}{2}$

Step 4.5.3

Simplify $\frac{65546 \pm 1536 \sqrt{1821}}{2}$ .

$p = 32773 \pm 768 \sqrt{1821}$

Step 4.6

The final answer is the combination of both solutions.

$p = 32773 + 768 \sqrt{1821}, 32773 - 768 \sqrt{1821}$

Step 5

Exclude the solutions that do not make $p - 4^{4} \sqrt{p} - 5 = 0$ true.

$p = 32773 + 768 \sqrt{1821}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$p = 32773 + 768 \sqrt{1821}$

Decimal Form:

$p = 65545.99961858 \dots$