Problem 70 - Algebra and Trigonometry

Question

\(47-70\) The given equation involves a power of the variable. Find all real solutions of the equation. $$ 6 x^{2 / 3}-216=0 $$

Step-by-Step Solution

Answer

The real solution is \(x = 216\).

Step 1: Isolate the Power Expression

First, we isolate the term with the power by adding 216 to both sides of the equation:\[ 6x^{2/3} - 216 = 0 \]\[ 6x^{2/3} = 216 \]

Step 2: Simplify the Coefficient

Next, divide both sides of the equation by 6 to simplify the expression:\[ x^{2/3} = \frac{216}{6} \]\[ x^{2/3} = 36 \]

Step 3: Remove the Fractional Exponent

To solve for \(x\), raise both sides of the equation to the power that is the reciprocal of \( \frac{2}{3} \), which is \( \frac{3}{2} \):\[ (x^{2/3})^{3/2} = 36^{3/2} \]\[ x = 36^{3/2} \]

Step 4: Calculate the Result

Calculate \(36^{3/2}\). Start by finding the square root of 36, which is 6, and then raise it to the power of 3:\[ 36^{3/2} = (36^{1/2})^3 = 6^3 = 216 \]Thus, \(x = 216\).

Key Concepts

Fractional Exponents

Fractional exponents are a unique way to represent roots in mathematical expressions. They can seem intimidating at first, but once broken down, they are simple to understand.

• The denominator of the fractional exponent represents the root.
• The numerator represents the power to which that root is raised.

For example, in the expression \(x^{2/3}\), the denominator 3 signifies the cube root. The numerator 2 indicates that the cube root is squared. Thus, \(x^{2/3}\) is equivalent to \(\sqrt[3]{x^2}\).
Knowing how to manipulate these exponents is crucial for solving equations involving roots and powers. Always remember to perform the operation in the correct order, dealing with the root first and then the power.

Isolation of Variables

When solving equations, especially those involving fractional exponents, isolation of variables is an essential step. It means rearranging the equation so that the variable of interest is on one side.

• Start by removing any constants or coefficients that are added or subtracted from the term.
• Use inverse operations, like addition or subtraction, to isolate the variable terms.

For the equation \(6x^{2/3} = 216\), we isolated \(x^{2/3}\) by first adding 216 to both sides, removing subtraction. Break complex operations into simple steps, making it less likely to make mistakes and easier to find solutions.

Algebraic Simplification

Algebraic simplification involves reducing expressions to a more manageable form. This process is used to make solving equations easier and to see correlations between terms clearly.

• Cancel out any common factors between terms, if applicable.
• Properly rearrange the terms to simplify the equation.

In the problem \(x^{2/3} = \frac{216}{6}\), the simplification was dividing 216 by 6, reducing the fraction to 36, and making the computation of \(x\) more straightforward. Simplification reduces errors, and saves time, leading you directly to the solution.

Real Solutions

Real solutions of an equation are the actual values of \(x\) that satisfy the equation within the real number system. Unlike imaginary or complex solutions, they are numbers that can be plotted on a traditional number line.

• Check the original problem conditions to ensure the solutions found are valid.
• Ensure operations respect the domain of the original functions involved.

In this exercise, we solved \(x^{2/3} = 36\) to find the real solution \(x = 216\). Testing your solution in the original equation helps confirm its validity within the real number scope. Finding real solutions means checking each step closely and ensuring calculations remain within real number properties.