Problem 6 Solve each equation. $$y^{3}=1... [FREE SOLUTION]

Chapter 7: Problem 6

Solve each equation. $$y^{3}=125$$

Short Answer

Expert verified

The solution is $y = 5$.

Step by step solution

Understanding the Given Equation

The given equation is a cubic equation: $y^{3} = 125$. This means we need to find a value of $y$ such that when it is raised to the power of 3, it results in 125.

Taking the Cube Root

To find $y$, we need to take the cube root of both sides of the equation. The cube root of $y^{3}$ is $y$, and the cube root of 125 is 5. Therefore, we can write: \[y = \sqrt[3]{125} \ = 5\].

Verify the Solution

To verify the solution, substitute $y = 5$ back into the original equation: $y^{3} = 5^{3} = 125$. Since both sides are equal, the solution $y = 5$ is correct.

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Key Concepts

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

cube root

To solve the cubic equation, we often need to find the cube root of a number. The cube root of a number is a value that, when multiplied by itself three times, gives the original number. For instance, the cube root of 125 is 5 because 5 multiplied by itself three times (5 脳 5 脳 5) equals 125.
This can be represented mathematically as $\[ \text{y}^3 = 125 \rightarrow y = \sqrt[3]{125} = 5 \]$.
Understanding cube roots is crucial in solving cubic equations. They help simplify complex expressions and find exact solutions.

verifying solutions

After solving any equation, it鈥檚 essential to verify the solution by substituting it back into the original equation. This ensures the solution is correct. For the equation in our exercise, we found that $\text{y} = 5$.
To verify, substitute $\text{y} = 5$ into the equation:

\[5^3 = 125 \]\
Since both sides of the equation are equal, the solution is correct.
Verification instills confidence in our solution and ensures we haven't made a mistake during the calculation process.

cubic equations

Cubic equations are polynomial equations of degree three. They often take the form $\text{ax}^3 + \text{bx}^2 + \text{cx} + \text{d} = 0$, where $\text{a} e 0$.
In our exercise, the cubic equation is simpler and looks like this: $\text{y}^3 = 125$. To solve cubic equations, we sometimes use techniques like:

Factoring
Using the cubic formula
Taking cube roots (as demonstrated)

Understanding these techniques is important for solving more complex cubic equations and for various applications in algebra, physics, and engineering.

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91影视

Solve each equation. $$y^{3}=125$$

Short Answer

Step by step solution

Understanding the Given Equation

Taking the Cube Root

Verify the Solution

Key Concepts

cube root

verifying solutions

cubic equations

One App. One Place for Learning.

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