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Chapter 10: Problem 24

Solve. $$\sqrt[3]{y}=-4$$

Short Answer

Expert verified
y = -64

Step by step solution

01

Understand the problem

We are given the equation \(\root{3}\root{y} = -4\). The goal is to solve for \(y\).
02

Isolate the cube root

In this problem, the cube root is already isolated. We have \( \root{3}\root{y} = -4 \).
03

Cube both sides

To eliminate the cube root, we will cube both sides of the equation: \[ (\root{3}\root{y})^3 = (-4)^3 \]
04

Simplify both sides

Cubing the cube root of \(y\) gives us \(y\), and cubing \(-4\) gives us \(-64\): \[ y = -64 \]

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Key Concepts

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

solving equations
Solving equations involves finding the value of a variable that makes the equation true. Let's break this down using our original problem, \(\sqrt[3]{y} = -4\). Our goal is to solve for the variable \(y\). In this instance, this means isolating and manipulating mathematical expressions to find the value that satisfies the equation. This can involve various operations such as addition, subtraction, multiplication, division, and as we'll see, cubing both sides. It's important to understand that each step in solving equations should be performed carefully to ensure accuracy, and the same operation must be done to both sides of the equation in order to maintain equality.
cubing both sides
To solve the given cubed root equation, \(\sqrt[3]{y} = -4\), we need to get rid of the cube root. One effective method is to cube both sides of the equation. Cubing both sides means raising each side of the equation to the power of three, which cancels out the cube root. This is because the cube of a cube root of a number is the number itself. Mathematically, this step looks like:
\[ \left(\sqrt[3]{y}\right)^3 = (-4)^3 \]
When we cube the left side, the cube root and the cube cancel each other, leaving us with just \(y\). On the right side, we cube -4, resulting in -64. This transformation simplifies our equation significantly.
simplifying equations
After cubing both sides of the equation \(\sqrt[3]{y} = -4\), we simplify the resulting expressions. Simplification is the process of reducing an equation or expression to its simplest form. In this case, after cubing the left side, we get \(y\), because \(\left(\sqrt[3]{y}\right)^3 = y\). On the right side, we calculate \(\left(-4\right)^3\) which equals -64. Therefore, our equation simplifies to:

\[ y = -64 \]
The final simplification tells us the value of \(y\) that satisfies the original equation. Simplification helps make the equation easier to work with and understand. It's the final, crucial step that leads us to the solution of the problem.

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