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Chapter 7: Problem 65

\(\sqrt[3]{r^{2}+2 r+8}=\sqrt[3]{r^{2}+3 r+12}\)

Short Answer

Expert verified
r=-4.

Step by step solution

01

- Equate Radicands

Since the cube roots of the expressions are equal, we can equate the radicands directly: r^{2}+2r+8=r^{2}+3r+12.
02

- Simplify the Equation

Subtracting r^2 from both sides: 2r+8=3r+12.
03

- Isolate the Variable

Subtract 2r and 12 from both sides: -4=r.

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Key Concepts

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

Equating Radicands
When you have an equation that involves cube roots on both sides, you can equate the radicands. In simple terms, the expressions inside the cube root must be equal for the equation to hold true.
For example, in the exercise: \(\sqrt[3]{r^{2}+2 r+8} = \sqrt[3]{r^{2}+3 r+12}\), you can set the expressions inside the cube roots equal to each other: \(r^{2} + 2r + 8 = r^{2} + 3r + 12\).

This step eliminates the cube roots and makes the problem much easier to solve.

    You are left with a simpler algebraic equation: \(r^{2} + 2r + 8 = r^{2} + 3r + 12\). This is thanks to the property that if \(\sqrt[3]{a} =\sqrt[3]{b}\), then \(a = b\).

Always remember to equate the radicands first, especially when dealing with equations involving cube roots.
Isolating Variables
Once you have equated the radicands and simplified your equation, the next step is to isolate the variable.
In our example, we start with: \(r^{2} + 2r + 8 = r^{2} + 3r + 12\).

The first move is to eliminate \(r^{2}\) from both sides to simplify the equation:
$$2r + 8 = 3r + 12$$

Next, rearrange the equation to get all the terms involving the variable \(r\) on one side and constants on the other side.
In this case, subtract \(2r\) and 12 from both sides:
$$2r + 8 - 2r - 12 = 3r + 12 - 2r - 12$$
This simplifies to: $$-4 = r$$

By isolating the variable in small, manageable steps, the equation becomes much easier to solve.
Simplifying Algebraic Equations
Simplifying involves reducing the equation to its most basic form. This is often done through a combination of addition, subtraction, multiplication, or division.
In our specific example, once we equated the radicands and isolated the variable, we had a simpler equation:
  • First, we simplified by eliminating \(r^{2}\)
    $$2r + 8 = 3r + 12$$
  • Next, further simplified by isolating the variable:
    $$- 4 = r$$


It's essential to follow these steps gradually:
  • Combine like terms: Gather all like variables and constants on one side.
  • Keep track of signs: Ensure you correctly add or subtract terms.
  • Simplify wherever possible: Reducethe expression to its simplest form.


In this way, you can easily solve algebraic equations and understand the process better.

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