Problem 74 - Algebra and Trigonometry

Question

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 Term

First, we need to isolate the term with the power. The equation given is \(6x^{2/3} - 216 = 0\). To isolate \(x^{2/3}\), add 216 to both sides:\[6x^{2/3} = 216\]

Step 2: Divide to Solve for the Power Term

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

Step 3: Undo the Fractional Exponent

To solve for \(x\), we need to undo the fractional exponent \(\frac{2}{3}\). This is done by raising both sides to the reciprocal power \(\frac{3}{2}\):\[(x^{2/3})^{3/2} = 36^{3/2}\]The left side simplifies to \(x\), while the right side requires calculation. Since 36 is a perfect square, find the square root and then cube it:\[36^{3/2} = (36^{1/2})^3 = 6^3 = 216\]

Step 4: Verify the Solution

Verify the solution by substituting \(x = 216\) back into the original equation:\[6(216^{2/3}) - 216\]Calculate \(216^{2/3}\):\[216^{2/3} = (6^3)^{2/3} = 6^2 = 36\]Thus:\[6 \times 36 - 216 = 216 - 216 = 0\]The original equation holds true, verifying that \(x = 216\) is a correct solution.

Key Concepts

Fractional Exponents

Fractional exponents, also known as rational exponents, represent roots and powers of numbers simultaneously. Understanding fractional exponents is crucial for solving equations like \(6x^{2/3} - 216 = 0\). A fractional exponent like \(x^{2/3}\) indicates that the base \(x\) is raised to the power of \(2\) and then the cube root is taken.To interpret fractional exponents:

• The numerator of the fraction indicates the power to which the base is raised.
• The denominator of the fraction indicates the root to be taken.

For instance, in \(36^{3/2}\), the base 36 is first square-rooted (because of the 2 in the denominator), and then cubed (because of the 3 in the numerator). This method simplifies complex calculations and helps when solving equations with powers and roots. Breaking down each step it helps strengthen the understanding of handling these kinds of expressions.

Real Solutions

In the context of solving equations, a real solution is a number that satisfies the equation within the set of real numbers. The problem involves finding real solutions for the given equation \(6x^{2/3} - 216 = 0\). Real numbers include all the numbers you can find on the number line - both positive and negative integers, fractions, and irrational numbers.To determine real solutions:

• Ensure that the transformations and operations on the equation do not exclude any valid real numbers.
• Verify by substitution to ensure that the solution satisfies the original equation.

Finding the real solutions involves validating that the calculation balances both sides of the equation when the solution is substituted back, as shown in the exercise where \(x = 216\) solves the equation.

Isolate the Variable

The first step in solving any algebraic equation is often to isolate the variable of interest. Isolation means rewriting the equation so that the variable stands alone on one side. In our problem \(6x^{2/3} - 216 = 0\), we isolate \(x^{2/3}\) to make solving easier.To isolate the variable:

• First, move constant terms to the opposite side of the equation by adding or subtracting them from both sides.
• Then, if the variable is part of a product, divide the entire equation by the coefficient of the variable term.

Isolating the variable is crucial because it allows you to focus on solving for the unknown. Once isolated, further steps like taking roots or raising to powers (as needed when dealing with fractional exponents) become straightforward.