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Chapter 10: Problem 23

Solve. $$\sqrt[3]{x}=-3$$

Short Answer

Expert verified
x = -27

Step by step solution

01

Understand the Equation

The equation given is \text{cube root of } x = -3. This can be written as \ \ \[ \ \sqrt[3]{x} = -3 \ \ \]
02

Eliminate the Cube Root

To solve for \(x\), we need to eliminate the cube root by cubing both sides of the equation. This gives us the equation: \ \ \[ \ \left( \sqrt[3]{x} \right)^{3} = (-3)^{3} \ \ \]
03

Simplify Both Sides

Simplifying both sides, we get \ \ \[ \ x = -27 \ \ \]
04

Verify the Solution

To ensure correctness, substitute \(x = -27\) back into the original equation and verify: \ \ \[ \ \ \sqrt[3]{-27} = -3 \ \ \]As both sides of the equation are equal, the solution is correct.

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Key Concepts

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

solving cube root equations
Cube root equations involve finding a value that, when raised to the power of three, equals a given number. In the exercise above, we're given the equation \(\root 3 \/{x} = -3\).
To solve cube root equations, follow these steps:
  • Read and understand the equation.
  • Identify that the equation contains a cube root.
  • Recognize what operation needs to be performed to isolate the variable.
For cube root equations, you need to eliminate the cube root to solve for the variable. This leads us to the next important step in solving these equations.
eliminating the cube root
Eliminating the cube root is essential to finding the solution to a cube root equation. Here are the steps:
  • Isolate the cube root expression if necessary.
  • Cube both sides of the equation to remove the cube root.
Applying this to our exercise:
Starting with \(\root 3 \/{x} = -3\)
Cube both sides: \(({ \root 3 \/{x}} )^{3} = (-3)^{3}\)
This simplifies to \ x = -27\
Cubing both sides is crucial because it transforms the equation into a simpler, polynomial form that can be more easily solved. Remember to perform the cubing operation carefully to ensure accuracy.
verifying solutions
Verification is an important step in solving equations, ensuring that the solution is correct.
To verify a solution for a cube root equation:
  • Substitute the solution back into the original equation.
  • Check if both sides of the equation are equal using this substituted value.
Let's verify our solution, \ x = -27\:
Substitute \ x = -27\ back into the original equation \(\root 3 \/{x} = -3\)
This gives us \ \root 3 \/{-27} = -3 \
The cube root of -27 is indeed -3, confirming that our solution is correct. Verification helps to catch any mistakes and confirms that the solution satisfies the original equation.

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