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Chapter 8: Problem 46

Solve equation. \(\sqrt[3]{r+1}+1=0\)

Short Answer

Expert verified
The solution is \(r = -2\).

Step by step solution

01

- Isolate the Cube Root

Move the constant term to the other side of the equation: \ \( \ \sqrt[3]{r+1} = -1 \)
02

- Cube Both Sides

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

- Solve for r

Subtract 1 from both sides to solve for \(r\): \ \( r = -1 - 1 \) \ Therefore: \ \( r = -2 \)

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Key Concepts

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

solving equations
Solving equations is a fundamental aspect of algebra. The goal is to find the value of the variable that makes the equation true. Here, we are given the equation \(\sqrt[3]{r+1}+1=0\). To solve it, we need to isolate the variable \(r\). This involves a series of steps:
  • First, isolate the cube root term. Move the constant to the other side of the equation: \( ail y{\sqrt[3]{r+1} = -1}\).
  • Second, eliminate the cube root by cubing both sides of the equation: \((\sqrt[3]{r+1})^3 = (-1)^3\). This simplifies to \(r + 1 = -1\).
  • Finally, solve for \(r\) by subtracting 1 from both sides: \(r = -1 - 1\), which results in \(r = -2\).
This step-by-step process is key to solving the given problem effectively.
algebraic manipulation
Algebraic manipulation means rearranging and simplifying equations using algebraic rules. It helps to bring the equation in a form where the solution becomes clear. In our example, the important steps of algebraic manipulation include:
  • Shifting the constant term to the other side: \(\sqrt[3]{r+1} = -1\).
  • Removing the cube root by cubing both sides: \((\sqrt[3]{r+1})^3 = (-1)^3\).
  • Solving the simplified equation: \(r + 1 = -1\).Subtract 1 from both sides to find \(r = -2\).
These manipulations help in isolating the variable and finding the correct value of \(r\). Breaking the problem into these smaller steps makes it less intimidating and easier to understand.
cube roots
A cube root of a number is a value that, when cubed, gives the original number. The cube root operation is the inverse of cubing. In this equation, the cube root of \(r + 1\)needs to be eliminated:
  • Start with \(\sqrt[3]{r+1} + 1 = 0\).Subtract 1 from both sides to isolate the cube root: \(\sqrt[3]{r+1} = -1\).
  • To get rid of the cube root, cube both sides: \( (\sqrt[3]{r+1})^3 = (-1)^3\).This results in: \(r + 1 = -1\).
  • Solve the resulting linear equation: Subtract 1 from both sides to find \(r = -2\).
Understanding how to work with cube roots is crucial for solving equations that involve them. Breaking down the operations step-by-step ensures clarity and makes solving such problems more approachable.

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