Question

In Exercises 3–12, solve the equation. Check your solution. $$ \sqrt[3]{x}-10=-7 $$

Short answer

The solution of the equation is \(x = 27\).

Step-by-step solution

Simplify the equation

First, isolate the cubic root by moving the -10 to the other side of the equation. This is done by adding 10 to both sides of the equation, resulting in \( \sqrt[3]{x} = -7+10 \).

Further simplify the equation

Simplifying the right side of the equation gives the equation \( \sqrt[3]{x} = 3 \).

Solve for \( x \)

To solve for \( x \), remove the cubic root by taking both sides to the power of 3, giving the equation \( x = 3^3 \).

Find the value of \( x \)

Solving \ (3^3)\, we get the solution \( x = 27 \).

Key concepts

Equation Solving

Solving equations is about finding unknown values that make the equation true. In this exercise, we need to solve a cubed root equation. The process typically involves isolating the variable on one side of the equation. We start with the given equation, \( \sqrt[3]{x} - 10 = -7 \).
The first step in solving the equation is to move all constants to one side. For our equation, we can do this by adding 10 to both sides to isolate the cubed root term. This transforms the equation into \( \sqrt[3]{x} = -7 + 10 \). Simplifying gives us \( \sqrt[3]{x} = 3 \).
This systematic approach of isolating terms helps simplify the equation and makes it manageable to find the unknown value. Ensuring that the variable is on one side of the equation is an essential part of the equation-solving process.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to simplify or solve them. Here, we manipulate the equation by rearranging terms to isolate the variable. Starting from \( \sqrt[3]{x} - 10 = -7 \), we want to make the root term \( \sqrt[3]{x} \) stand alone on one side, as seen in the previous section.

• Firstly, add 10 to both sides. This is because we need the \( -10 \) on one side of the equation to move it to the other side.
• This leads to \( \sqrt[3]{x} = 3 \). This represents our next step.

We now have a simpler equation where the cubed root is isolated. Getting this single step right is crucial as it affects how we proceed with solving for \( x \).
These algebraic manipulations help to reduce complexities in equations and make solving possible without errors.

Cubic Roots

Cubic roots are the opposite of cubing a number. They ask what number, when multiplied by itself three times, equals the given value. In our equation, we're looking to find \( x \) such that \( x^3 = 3^3 \).
The important point here is to understand that when you remove a cubic root, you do so by cubing the entire expression. So to solve \( \sqrt[3]{x} = 3 \) for \( x \), you cube both sides of the equation. This gives \( x = 3^3 \).

• Cubing both sides is important for preserving the balance of the equation.
• Through cubing, you effectively "cancel out" the cube root on one side.

Finally, we calculate \( 3^3 \), resulting in \( 27 \). Thus, \( x = 27 \) is the solution. Understanding cubic roots is crucial as they frequently appear in algebra and can initially seem daunting. However, following methodical steps like these, you can easily tackle equations that involve cubic roots.