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Chapter 5: Problem 16

Solve each equation by hand. Do not use a calculator. $$\sqrt[3]{x+9}=2$$

Short Answer

Expert verified
The solution is x = -1.

Step by step solution

01

Eliminate the Cube Root

To solve the equation with a cube root, we start by eliminating the cube root by raising both sides of the equation to the power of 3. This will give:\[(\sqrt[3]{x+9})^3 = 2^3\]Simplifying both sides gives:\[x + 9 = 8\]
02

Isolate the Variable x

Now we need to isolate the variable \(x\). To do this, subtract 9 from both sides of the equation:\[x + 9 - 9 = 8 - 9\]This simplifies to:\[x = -1\]

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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 all about finding the value of the variable that makes the equation true. In this context, we start with an equation that involves a cube root. The main goal here is to simplify the equation step-by-step until we can easily determine the value of the variable. With practice, this process becomes quicker and more intuitive. It typically involves tweaking one side of the equation, so it matches the other. When solving equations:
  • Identify terms containing the variable.
  • Make strategic transformations to isolate the variable.
  • Perform operations evenly on both sides to maintain balance.
The process requires patience and attention to detail, ensuring each step logically and correctly follows the previous one.
Isolating Variables
To isolate a variable means to get it alone on one side of the equation, with everything else on the opposite side. In this exercise, after eliminating the cube root, we're left with the expression \( x + 9 = 8 \). Here comes the importance of basic arithmetic operations. By subtracting 9 from both sides, we remove the constant that is with the variable. This step is crucial for isolation.Key points to remember when isolating variables:
  • Always do the same operation on both sides of the equation.
  • Use the inverse operation to remove constants or coefficients. For example, if a number is added to \(x\), subtract it.
  • Simplify incrementally, ensuring the variable stands alone.
Once isolated, you have effectively solved the equation, revealing the value of the variable.
Exponentiation
Exponentiation is the opposite of taking roots. To remove a cube root, we cube the expression. In this exercise, we raised both sides of the equation \( \sqrt[3]{x+9} = 2 \) to the power of 3. If you cube a cube root, they cancel each other out, revealing the hidden value:\[ (\sqrt[3]{x+9})^3 = 2^3 \rightarrow x + 9 = 8 \]It's crucial to apply the exponentiation correctly, ensuring it affects the whole side of the equation. This practice guarantees the cancellation of the cube root effectively.Things to remember about exponentiation:
  • It neutralizes the root operation; cubing a cube root leaves only the inner term.
  • Apply it uniformly across both sides of the equation to maintain equality.
  • It helps in revealing the expression or constant hidden within the root.
Through careful use of this mathematical operation, the equation becomes simpler, leading us closer to the solution.

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Most popular questions from this chapter

Solve each problem. Volume of a Cylinder The volume of a right circular cylinder is jointly proportional to the square of the radius of the circular base and to the height. If the volume is 300 cubic centimeters when the height is 10.62 centimeters and the radius is 3 centimeters, approximate the volume of a cylinder with radius 4 centimeters and height 15.92 centimeters. (image can't copy)

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