Solve for p (p^2*p^4)^(1/3)+5=13

${(p^{2} \cdot p^{4})}^{\frac{1}{3}} + 5 = 13$

Step 1

Move the terms containing $p$ to the left side and simplify.

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

Subtract $5$ from both sides of the equation.

${(p^{2} \cdot p^{4})}^{\frac{1}{3}} = 13 - 5$

Subtract $5$ from $13$ .

${(p^{2} \cdot p^{4})}^{\frac{1}{3}} = 8$

Multiply $p^{2}$ by $p^{4}$ by adding the exponents.

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

${(p^{2 + 4})}^{\frac{1}{3}} = 8$

Add $2$ and $4$ .

${(p^{6})}^{\frac{1}{3}} = 8$

Step 2

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${({(p^{6})}^{\frac{1}{3}})}^{3} = 8^{3}$

Step 3

Simplify the exponent.

Simplify the left side.

Simplify ${({(p^{6})}^{\frac{1}{3}})}^{3}$ .

Multiply the exponents in ${({(p^{6})}^{\frac{1}{3}})}^{3}$ .

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

${(p^{6})}^{\frac{1}{3} \cdot 3} = 8^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(p^{6})}^{\frac{1}{3} \cdot 3} = 8^{3}$

Rewrite the expression.

${(p^{6})}^{1} = 8^{3}$

Multiply the exponents in ${(p^{6})}^{1}$ .

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

$p^{6 \cdot 1} = 8^{3}$

Multiply $6$ by $1$ .

$p^{6} = 8^{3}$

Simplify the right side.

Raise $8$ to the power of $3$ .

$p^{6} = 512$

Step 4

Solve for $p$ .

Take the 6th root of both sides of the equation to eliminate the exponent on the left side.

$p = \pm \sqrt[6]{512}$

Simplify $\pm \sqrt[6]{512}$ .

Rewrite $512$ as $2^{6} \cdot 8$ .

Factor $64$ out of $512$ .

$p = \pm \sqrt[6]{64 (8)}$

Rewrite $64$ as $2^{6}$ .

$p = \pm \sqrt[6]{2^{6} \cdot 8}$

Pull terms out from under the radical.

$p = \pm 2 \sqrt[6]{8}$

Rewrite $8$ as $2^{3}$ .

$p = \pm 2 \sqrt[6]{2^{3}}$

Rewrite $\sqrt[6]{2^{3}}$ as $\sqrt{\sqrt[3]{2^{3}}}$ .

$p = \pm 2 \sqrt{\sqrt[3]{2^{3}}}$

Pull terms out from under the radical, assuming real numbers.

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

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

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

$p = 2 \sqrt{2}$

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

$p = - 2 \sqrt{2}$

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

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$p = 2.82842712 \dots, - 2.82842712 \dots$

Solve for p (p^2*p^4)^(1/3)+5=13

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