Problem 72 - Algebra and Trigonometry

Question

\(71-74\) . Solve the equation for the variable \(x\) . The constants \(a\) and \(b\) represent positive real numbers. $$ a^{3} x^{3}+b^{3}=0 $$

Step-by-Step Solution

Answer

\(x = -\frac{b}{a}\)

Step 1: Start with the Original Equation

The original equation is given by \(a^3 x^3 + b^3 = 0\). We need to isolate \(x\).

Step 2: Isolate the Cubic Term

Move the term \(b^3\) to the other side of the equation. We get:\[a^3 x^3 = -b^3\]

Step 3: Divide Both Sides by the Cube of a

Divide both sides of the equation by \(a^3\) to solve for \(x^3\) as follows:\[x^3 = -\frac{b^3}{a^3}\]

Step 4: Take the Cube Root

To solve for \(x\), take the cube root of both sides of the equation:\[x = \sqrt[3]{-\frac{b^3}{a^3}}\]

Step 5: Simplify the Expression

Simplify the expression using the property of cube roots:\[x = -\frac{b}{a}\]

Key Concepts

Understanding a Cubic Equation

A cubic equation is any equation that involves a variable raised to the third power. In general, it can be expressed in the form: \(ax^3 + bx^2 + cx + d = 0\). In the context of this exercise, our cubic equation is relatively simpler: \(a^3 x^3 + b^3 = 0\). Cubic equations can have several characteristics:

• They can have up to three real roots.
• The terms involve powers of the variable up to three.
• Solving these equations often involves isolating the cubic term.

By understanding the nature of these equations, you can better grasp each step required for isolating and solving for the variable.

Method to Isolate the Variable

Isolating the variable means rearranging the equation such that the variable is on one side, and everything else is on the other. This allows us to solve directly for the variable.In our equation \(a^3 x^3 + b^3 = 0\), we need to make \(x\) the subject by performing the following steps:

• First, move \(b^3\) to the right side of the equation, which isolates the cubic term: \(a^3 x^3 = -b^3\).
• Second, divide each side by \(a^3\) to focus on the term \(x^3\): \(x^3 = -\frac{b^3}{a^3}\).

These actions allow us to simplify and prepare the equation for further operations such as taking the cube root.

Working with Positive Real Numbers

In mathematics, real numbers include all the rational and irrational numbers. Positive real numbers are any real numbers greater than zero. In our problem, constants \(a\) and \(b\) represent positive real numbers, which means:

• They are greater than zero.
• The calculation and operations using these constants follow regular algebraic rules.

Their positivity ensures that operations like division remain valid and do not result in undefined or complex numbers, which simplifies solving the equation. This groundwork is crucial for maintaining straightforward operations like dealing with roots.

Understanding and Using the Cube Root

The cube root of a number is the value that, when multiplied by itself three times, gives the original number. The cube root is denoted through notation like \(\sqrt[3]{...}\) or raising to the power of \(1/3\).For the equation \(x^3 = -\frac{b^3}{a^3}\), taking the cube root on both sides results in: \(x = \sqrt[3]{-\frac{b^3}{a^3}}\). Simplifying further, given the symmetry of the cube root operation, results in: \(x = -\frac{b}{a}\).Key things to remember about cube roots:

• They differ from square roots, as they allow for negative numbers.
• Remembering cube root properties simplifies equations.
• Simplification often leads directly to solutions.

Understanding cube roots is key to learning how to solve these cubic equations efficiently and with confidence.