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Chapter 7: Problem 16

$$ y^{3}=-27 $$

Short Answer

Expert verified
y = -3

Step by step solution

01

Isolate the Variable

The given equation is already simplified: \( y^3 = -27 \).So, the first step is to eliminate the power of 3 by taking the cube root of both sides.
02

Take the Cube Root

To solve for \( y \), take the cube root of both sides. \( y = \root 3 \fr{-27} \). Cube root of -27 is -3.
03

Solution

Thus, \( y = -3 \).

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Key Concepts

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

Cube Root
Taking the cube root of a number means finding a value that, when multiplied by itself three times, will give the original number.
For example, since \(3 \times 3 \times 3 = 27\), the cube root of 27 is 3, and since \(-3 \times -3 \times -3 = -27\), the cube root of -27 is -3.
To denote the cube root of a number, we use the symbol \( \root 3 \). So, \( \root 3 \fr{-27} = -3 \).
This operation helps in solving cubic equations by simplifying them into more manageable forms.
Whenever you encounter a cube root, remember it only has one real solution, unlike square roots that can have both positive and negative roots.
Isolating Variables
When solving equations, isolating the variable helps simplify the problem. It involves manipulating the equation until the variable you want to solve for (in this case, \(y\)) is on one side of the equation by itself.

Initially, our equation was:
\(y^{3} = -27\)
Since \(y\) is already on one side of the equation, it means we have isolated the variable from the start.
Now, to isolate \(y\) completely, we need to undo the cube by taking the cube root of both sides of the equation.
This leaves us with: \( y = \root 3 \fr{-27} \)
Negative numbers
Negative numbers are numbers less than zero, usually represented with a minus sign (-).
When dealing with cubic equations and cube roots, understanding how negative numbers operate is crucial.
For example, multiplying a negative number three times results in a negative product:
  • (-3) 脳 (-3) 脳 (-3) = -27
      • This property is why the cube root of -27 is negative (-3), unlike square roots, which do not have real negative roots because a negative number squared results in a positive value.
      Remember:
      - The product of an odd number of negative factors will always be negative.
      - The product of an even number of negative factors will be positive.

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Most popular questions from this chapter

Determine whether the values in each table could represent a linear relationship, a quadratic relationship, or neither. Explain your answers. $$ \begin{array}{|c|c|c|c|c|c|c|c|}\hline x & {-3} & {-2} & {-1} & {0} & {1} & {2} & {3} \\ \hline y & {-12.6} & {-9.2} & {-5.8} & {-2.4} & {1} & {4.4} & {7.8} \\ \hline\end{array} $$

Solve each equation by completing the square. $$ 2 u^{2}+3 u-2=0 $$

Factor each quadratic expression that can be factored using integers. Identify those that cannot, and explain why they can't be factored. $$ z^{2}+2 z-6 $$

Solve each equation by factoring using integers, if possible. If an equation can't be solved in this way, explain why. $$ 4 x+x^{2}=21 $$

Factor the expression on the left side of each equation as much as possible, and find all the possible solutions. It will help to remember that \(x^{4}=\left(x^{2}\right)^{2}, x^{8}=\left(x^{4}\right)^{2},\) and \(x^{3}=x\left(x^{2}\right) .\) $$ x^{4}-1=0 $$
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