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Chapter 10: Problem 46

Solve each equation. $$ \sqrt[3]{r+1}+1=0 $$

Short Answer

Expert verified
r = -2

Step by step solution

01

Isolate the Cubed Root Term

Begin by isolating the cubed root term in the equation \ \ \( \sqrt[3]{r+1}+1=0 \ \ \). Subtract 1 from both sides of the equation: \ \ \( \sqrt[3]{r+1} = -1 \ \ \).
02

Cube Both Sides

To eliminate the cubed root, cube both sides of the equation: \ \ \( \left( \sqrt[3]{r+1} \right)^3 = (-1)^3 \ \ \). This simplifies to: \ \ \( r+1 = -1 \ \ \).
03

Solve for r

Finally, solve for \( r \) by subtracting 1 from both sides of the equation: \ \ \( r+1-1 = -1-1 \ \ \). This simplifies to: \ \ \( r = -2 \ \ \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Cubed Root
The cubed root operation is the reverse of cubing a number. It helps us figure out what number, when multiplied by itself three times, gives us the original number.
For example:
  • The cubed root of 27 is 3, because 3 * 3 * 3 = 27
  • The cubed root of -8 is -2, because -2 * -2 * -2 = -8
The symbol for the cubed root is \sqrt[3]{...}.
In our equation, $$\sqrt[3]{r+1}+1=0$$, we need to isolate the cubed root part to solve it.
Isolating Variables
Isolating the variable means getting the variable by itself on one side of the equation.
This involves using inverse operations to undo whatever is being done to the variable. Let's break it down:
In the equation \(\sqrt[3]{r+1}+1=0\), we start by isolating the cubed root: We subtract 1 from both sides to get: \(\sqrt[3]{r+1} = -1\)
Now the cubed root term is by itself, making it easier to solve for $$r$.
Simplifying Expressions
Simplifying expressions makes equations easier to solve. It means performing operations to combine like terms and reduce complexity.
In our problem: \(\sqrt[3]{r+1} = -1\)
We need to eliminate the cubed root by applying the inverse operation鈥攃ubing. Cubing both sides, we get:
\((\sqrt[3]{r+1})^3 = (-1)^3\)
This simplifies to:
r+1 = -1
Finally, solve for $$r$$ by subtracting 1 from both sides:
\r-1 = -2
So, $$r = -2$$.

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