Question

Solve the equations exactly. Use an inverse function when appropriate. $$ \sqrt{x^{3}-2}=5 $$

Short answer

Answer: The solution to the given equation is $x = 3$.

Step-by-step solution

Write down the equation

The given equation is: $$ \sqrt{x^3 - 2} = 5 $$

Square both sides

To remove the square root sign, square both sides of the equation: $$ (\sqrt{x^3 - 2})^2 = 5^2 $$ This simplifies to: $$ x^3 - 2 = 25 $$

Add 2 to both sides

Add 2 to both sides of the equation to isolate the term containing the variable x: $$ x^3 = 27 $$

Find the inverse function

Since the exponent is 3, the inverse function is the cube root. Apply the cube root to both sides of the equation: $$ \sqrt[3]{x^{3}} = \sqrt[3]{27} $$ This simplifies to: $$ x = 3 $$ So, the solution to the given equation is: $$ x = 3 $$

Key concepts

Solving Equations

Equations are mathematical statements that assert the equality of two expressions. Solving equations involves finding the value of the variable that makes the equation true. In our example, we started with \( \sqrt{x^3 - 2} = 5 \). The goal is to isolate the variable \(x\). This process often involves performing operations that simplify the equation, making it easier to solve.

In the first step, you need to identify the operations that can be reversed. Here, identifying that the square root needs to be removed is key. Reversing operations is often done by applying inverse operations. For instance, the square root is undone by squaring.

When solving equations, always maintain balance; anything done to one side of the equation should also be done to the other. This will help in keeping the equation valid while manipulating it to find the solution. Pay attention to detail to avoid mistakes that may throw off your answers.

Cube Root

The cube root is an inverse operation of cubing a number. If a number \(x\) is cubed, then \( \sqrt[3]{x} \) gives the value that when cubed results in \(x\). In the given solution, you end up with \(x^3 = 27\).

Here, applying the cube root to both sides gives \( \sqrt[3]{x^3} = \sqrt[3]{27} \). This ultimately simplifies to \(x = 3\), because \(3 \times 3 \times 3 = 27\).

Cube roots are particularly useful in solving equations where the variable is raised to the power of three. They provide a method to isolate the variable and find its value. Practice with different numbers to become more familiar with cube roots and their applications.

Square Root

The square root operation is used to find a number which, when multiplied by itself, results in the original number. In the equation \( \sqrt{x^3 - 2} = 5 \), we initially deal with the square root of \(x^3 - 2\).

To remove the square root in the equation, we perform the operation's inverse by squaring both sides. For example, squaring \( \sqrt{x^3 - 2} \) simplifies it to \(x^3 - 2\). Remember, squaring a square root doesn't just simplify — it eliminates the square root entirely.

Square roots are fundamental in tackling various algebra problems where the variable needs isolating. They serve as a crucial instrument in reversing square operations, ensuring you can work backward to find the variable's value. These operations reinforce the concept of inverse functions, where each action has a corresponding inverse that can be performed to obtain the original input.