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Chapter 10: Problem 38

Solve each equation. $$ \sqrt[3]{p+5}=\sqrt[3]{2 p-4} $$

Short Answer

Expert verified
p = 9

Step by step solution

01

Understand the Equation

The given equation is \( \sqrt[3]{p+5} = \sqrt[3]{2p-4} \). Both sides of the equation involve cube roots.
02

Remove the Cube Root

To solve for \( p \), we need to remove the cube roots by cubing both sides of the equation: \[ \left( \sqrt[3]{p+5} \right)^3 = \left( \sqrt[3]{2p-4} \right)^3 \] which simplifies to: \[ p+5 = 2p-4 \]
03

Isolate the Variable p

Rearrange the equation to isolate \( p \): \[ p + 5 = 2p - 4 \] Subtract \( p \) from both sides: \[ 5 = p - 4 \] Finally, add 4 to both sides: \[ p = 9 \]
04

Verify the Solution

Substitute \( p = 9 \) back into the original equation to verify: \[ \sqrt[3]{9 + 5} = \sqrt[3]{2(9) - 4} \] Simplifies to: \[ \sqrt[3]{14} = \sqrt[3]{14} \] Since both sides are equal, the solution is verified.

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