Question

Solve the equations exactly. Use an inverse function when appropriate. $$ \sqrt[3]{30-\sqrt{x}}=3 $$

Short answer

Answer: The value of x in the given equation is x = 9.

Step-by-step solution

Remove the cube root

To do this, we will cube both sides of the equation. $$ (\sqrt[3]{30-\sqrt{x}})^3 = 3^3 $$ Simplifying, we obtain: $$ 30 - \sqrt{x} = 27 $$

Isolate the square root

Add the value inside the cube root on both sides, effectively isolating the square root. $$ 30 - 27 = \sqrt{x} $$ Simplifying, we obtain: $$ 3 = \sqrt{x} $$

Solve for x

To find the value of x, square both sides of the equation. $$ (3)^2 = (\sqrt{x})^2 $$ Simplifying, we obtain: $$ 9 = x $$ The solution to the equation is x = 9.

Key concepts

Inverse Functions

In mathematics, inverse functions are a way of "undoing" the effect of a function. If you have a function that changes an input to an output, the inverse function takes that output back to the original input. They're particularly useful when solving equations because they allow us to reverse operations.

For example, if you have a function \( f(x) = 3x + 2 \), its inverse \( f^{-1}(x) \) would be the operation that takes the input back to the form before \( f \) was applied. In this exercise, the functions involved are root functions, and inversing means applying operations like cubing to reverse cube roots or squaring to reverse square roots.

Cube Root

The cube root of a number is a value that, when multiplied by itself three times, gives that number. For instance, the cube root of 8 is 2 because \( 2 \times 2 \times 2 = 8 \). In this exercise, you encounter the cube root function \( \sqrt[3]{x} \).

When solving the equation \( \sqrt[3]{30 - \sqrt{x}} = 3 \), applying the inverse, which is cubing both sides, effectively removes the cube root. This operation helps to simplify the equation and leads us toward solving it step by step. Cubing is crucial here to get rid of the cube root so we can deal directly with simpler algebraic expressions.

Square Root

A square root finds a number which, when multiplied by itself, will give the original number. For example, the square root of 9 is 3 because \( 3 \times 3 = 9 \).

In the exercise, after eliminating the cube root and simplifying to \( 30 - \sqrt{x} = 27 \), we isolate the square root function \( \sqrt{x} \) on one side, giving us \( 3 = \sqrt{x} \). To further simplify, we square both sides to eliminate the square root. This reversal operation of squaring is an example of using the inverse function to solve equations. It allows you to express \( x \) in its simplest form without any roots.

Isolating the Variable

Isolating the variable means rearranging an equation in such a way that one side of the equation has the variable we are solving for, and the other side has the known values. This process is essential in solving equations as it helps define the unknown value clearly.

In our exercise, we simplify by performing operations step-by-step:

• First, eliminate the cube root by cubing both sides.
• Next, manipulate the equation to isolate the square root, \( 3 = \sqrt{x} \).
• Finally, remove the square root by squaring both sides, which allows us to solve for \( x \).

Each step involves using algebraic manipulations to move terms around and simplify operations, ultimately leading to the solution \( x = 9 \). Correctly isolating the variable is critical, as shown in the logical progression of this problem-solving process.