Problem 36 - Intermediate Algebra

Question

Solve. \(\sqrt[3]{x-4}-5=-7\)

Step-by-Step Solution

Answer

The solution is \( x = -4 \).

Step 1: Isolate the Cube Root

Start by isolating the cube root expression on one side of the equation. To do this, add 5 to both sides of the equation: \( \sqrt[3]{x-4} - 5 + 5 = -7 + 5 \). This simplifies to \( \sqrt[3]{x-4} = -2 \).

Step 2: Eliminate the Cube Root

Next, remove the cube root by raising both sides of the equation to the power of 3 (since a cube root and a cube are inverse functions). Raise both sides: \( (\sqrt[3]{x-4})^3 = (-2)^3 \). This simplifies to \( x-4 = -8 \).

Step 3: Solve for x

Now, solve for \( x \) by adding 4 to both sides of the equation: \( x-4 + 4 = -8 + 4 \). This simplifies to \( x = -4 \).

Key Concepts

Cube Roots

Cube roots are the inverse operation of cubing a number. For example, the cube root of 8 is 2, because 2 cubed (or raised to the power of 3) equals 8. In algebra, cube roots are written as \( \sqrt[3]{x} \)

• The cube root of a number \( x \) is a value that, when multiplied by itself three times, gives \( x \).
• Cube roots can have negative results. For instance, \( \sqrt[3]{-8} = -2 \) because \( (-2)^3 = -8 \).
• Understanding cube roots is crucial in solving equations that involve them, as they allow us to "undo" cubing.

Recognizing when to use cube roots, especially in algebraic expressions, helps isolate and solve for variables effectively.

Isolating Variables

Isolating a variable is a fundamental step in solving equations. It involves manipulating the equation so that the unknown variable is alone on one side of the equation.

• Begin by eliminating any terms that are added or subtracted to the variable. This often involves simple arithmetic operations like adding or subtracting the same number from both sides of the equation.
• If the variable is part of a more complex expression, such as under a cube root, you'll need to carefully isolate that expression first before removing the cube root.
• The goal of isolating the variable is to simplify the equation as much as possible to find the variable's value.

In our example, isolating the cube root term \( \sqrt[3]{x-4} \) helped us simplify the equation, making it easier to solve.

Inverse Functions

Inverse functions are operations that "reverse" the effect of the original function. In the context of this exercise, the inverse of a cube root is cubing.

• An inverse function essentially undoes the action of a function. For the cube root function \( y = \sqrt[3]{x} \), the inverse function would be \( y^3 = x \).
• Using inverse functions is crucial when you need to eliminate certain operations to solve for a variable.
• Applying the cube to both sides of an equation effectively cancels out the cube root, letting us continue solving for the variable.

By understanding and using inverse functions, we can solve complex equations by breaking them down into simpler steps.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to simplify and solve them. It includes a variety of techniques that make equations easier to handle.

• Basic manipulation techniques include addition, subtraction, multiplication, and division.
• More complex manipulations might involve factoring, expanding expressions, or using properties of exponents and roots.
• The key is to perform equivalent operations on both sides of the equation to maintain equality.

In our example, careful algebraic manipulation was used every step of the way—first to isolate the cube root, then to use the inverse function, and finally to solve for the variable \( x \). These skills are essential in algebra and problem-solving in mathematics.