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Chapter 7: Problem 9

Solve each equation. $$(y-3)^{3}=-8$$

Short Answer

Expert verified
y = 1

Step by step solution

01

Isolate the expression inside the cube

Given the equation (y-3)^3 = -8, the expression inside the cube is already isolated.
02

Apply the cube root

To remove the cube, take the cube root of both sides of the equation: \( \sqrt[3]{ (y-3)^3 } = \sqrt[3]{ -8 } \) This simplifies to y-3 = -2.
03

Solve for y

Add 3 to both sides of the equation to isolate y: y - 3 + 3 = -2 + 3 This simplifies to y = 1.

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Key Concepts

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

cube root
In our problem, we encounter a cube root. A cube root answers the question: 'What number, when multiplied by itself three times, gives the original number?' For example, the cube root of -8 is -2 because (-2) 脳 (-2) 脳 (-2) = -8.
Taking the cube root is the reverse process of cubing a number. Cubed numbers grow faster because they involve three multiplications. Therefore, knowing how to take a cube root helps to solve equations where the variable is in a cube.
isolating variable
Isolating the variable is a common goal in algebraic equations. It means getting the variable alone on one side of the equation. This makes it possible to see the solution directly.
In our problem, the variable y is in the term (y - 3)^3. To isolate it, we first need to deal with the cube by taking the cube root of both sides. After that, subtracting or adding numbers helps to move all other terms to the opposite side.
This process simplifies the equation step by step until only the variable is left on one side.
solving algebraic equations
Solving algebraic equations involves finding the value of the variable that makes the equation true. Here's a structured way to approach it:
  • Identify the form of the equation and simplify if needed.
  • Isolate the variable by performing inverse operations.
  • Check each step to maintain equality.

In our exercise, we start with the equation (y - 3)^3 = -8. The goal is to find y:
  • First, identify that we need to take the cube root to simplify.
  • Next, solve for y by isolating the variable.
  • Finally, check the solution by substituting it back into the original equation.

This systematic approach ensures accuracy and helps grasping algebraic concepts efficiently.

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