Problem 72 - Algebra and Trigonometry

Question

The given equation involves a power of the variable. Find all real solutions of the equation. \(x^{4 / 3}-16=0\)

Step-by-Step Solution

Answer

The real solution is \(x = 8\).

Step 1: Isolate the Power

Begin by isolating the term containing the power. The given equation is \(x^{4/3} - 16 = 0\). Add 16 to both sides of the equation to get \(x^{4/3} = 16\).

Step 2: Convert Root to Exponent

The exponent \(4/3\) can be interpreted as \(x\) raised to the power \(4\), then taking the cube root. Therefore, we will solve for \(x\) by taking both sides to the power of \(3/4\): \(x = (16)^{3/4}\).

Step 3: Simplify the Exponent

First, find \(16^{1/4}\), which is the fourth root of 16. Since \(16 = 2^4\), the fourth root of 16 is \(2\). Then calculate \(2^3 = 8\). Thus, \(16^{3/4} = 8\).

Step 4: Verify the Solution

Substitute \(x = 8\) back into the original equation to ensure it satisfies it. Calculate \((8)^{4/3}\): The cube root of 64 (since \(8^2 = 64\)) is 4, which matches the rearranged part of our equation, confirming that the solution is correct.

Key Concepts

Equation Solving

Equation solving is a fundamental part of mathematics, where we find values for variables that make the equation true. In the given exercise, the equation is \(x^{4/3} - 16 = 0\). The goal is to find all the real values of \(x\) that satisfy this equation.

When solving equations, a common goal is to get the unknown variable, here \(x\), on one side by itself. This often involves rearranging the equation and performing the same operation on both sides to maintain equality. Initially, we have \(x^{4/3} - 16 = 0\). Adding 16 to both sides isolates the term with the variable, giving \(x^{4/3} = 16\). Once isolated, further steps focus on simplifying and solving for the variable.

• Start with isolating the variable
• Perform identical operations on both sides
• Simplify expressions where possible

Exponents and Roots

Exponents and roots are mathematical operations that are inverse to each other. An exponent indicates how many times a number, the base, is multiplied by itself. For example, \(x^{4/3}\) means that \(x\) is first raised to the fourth power and then the cube root is taken.

In the equation \(x^{4/3} = 16\), one goal is to eliminate the exponent to solve for \(x\). To do this, you can apply the reciprocal of the exponent as another exponent to both sides of the equation. Here, applying the \(3/4\) power isolates \(x\): \(x = (16)^{3/4}\).

Understanding how to manipulate exponents and roots is crucial. It involves

• Recognizing the relationship between the exponent and the operation needed to solve (in this case, taking reciprocals)
• Breaking down the process into manageable parts: solving first for the root and then the power

Isolation of Variables

Isolation of variables is a technique used in solving equations to make the variable of interest stand alone on one side of the equation. This often involves multiple operations and steps, particularly when the equation involves more complex forms like powers and roots.

In this particular problem, isolation began by adding 16 to both sides to remove any additional terms from the side of the equation containing the variable. Then, using operations involving exponents, we further isolated \(x\) by eliminating the fractional power. Solving\( x = (16)^{3/4} \) illustrates having completely isolated the variable.

The key to variable isolation includes:

• Carefully performing inverse operations to undo what has been applied to the variable
• Applying mathematical rules such as properties of exponents and roots
• Working through step-by-step manipulations

By following these principles, you’ll be more effective at addressing a variety of algebraic equations.