Problem 75 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation involving "nested" radicals for all real solutions analytically. Support your solutions with a graph. $$\sqrt[3]{\sqrt{32 x}}=\sqrt[3]{x+6}$$

Step-by-Step Solution

Answer

The real solution is \(x = 2\).

Step 1: Isolate the Cubed Roots

Rewrite the given equation to emphasize the cube roots:\[\sqrt[3]{\sqrt{32x}} = \sqrt[3]{x+6}\]Since both sides of the equation are cube roots, equate their radicands.

Step 2: Equate the Radicands

Since the cube roots are equal, their radicands must also be equal:\[\sqrt{32x} = x + 6\]This is the new equation that needs to be solved.

Step 3: Square Both Sides to Remove the Square Root

Square both sides of the equation to eliminate the square root:\[(\sqrt{32x})^2 = (x + 6)^2\]This simplifies to:\[32x = x^2 + 12x + 36\]

Step 4: Rearrange and Simplify the Quadratic Equation

Bring all terms to one side to form a standard quadratic equation:\[x^2 + 12x + 36 - 32x = 0\]This simplifies to:\[x^2 - 20x + 36 = 0\]

Step 5: Factor the Quadratic Equation

Factor the quadratic equation:\[x^2 - 20x + 36 = (x - 2)(x - 18) = 0\]This gives two potential solutions: \(x = 2\) and \(x = 18\).

Step 6: Verify the Solutions

Substitute each solution back into the original equation to verify their validity:- For \(x = 2\):\[\sqrt[3]{\sqrt{32 \times 2}} = \sqrt[3]{2 + 6}\]\[\sqrt[3]{\sqrt{64}} = \sqrt[3]{8}\]\[4^{1/3} = 2\]Valid, since both sides equal 2.- For \(x = 18\):\[\sqrt[3]{\sqrt{32 \times 18}} = \sqrt[3]{18 + 6}\]\[\sqrt[3]{\sqrt{576}} = \sqrt[3]{24}\]\[24^{1/3} eq 2\]Not valid, as the two sides are not equal. Thus, only \(x = 2\) is valid.

Key Concepts

Cube Roots

Cube roots help us find a number that, when multiplied by itself three times, gives an original number. This is very useful in algebra for simplifying complex expressions and solving equations. For example, the cube root of 8 is 2, because multiplying 2 three times gives 8: \(2 \times 2 \times 2 = 8\).

In this exercise, we work with cube roots of nested radicals, where a number is both under a cube root and another radical or power. Understanding how to simplify nested radicals by focusing on cube roots is crucial. By equating cube roots on both sides of an equation, we can simplify the problem to comparing simpler expressions.

Keep in mind that simplifying cube roots often helps to reduce complex equations to more manageable forms, making it easier to isolate variables or handle further calculations.

Quadratic Equation

Quadratic equations are of the form \(ax^2 + bx + c = 0\), with "a", "b", and "c" as coefficients, and "x" as the variable. These equations can be solved by various methods such as factoring, completing the square, or using the quadratic formula.

In our problem, after equating radicands and squaring both sides, we ended up with a quadratic equation: \(x^2 - 20x + 36 = 0\). Solving this equation leads us to potential solutions for "x" that can satisfy the original condition of equality.

Understanding the structure of quadratic equations and how to manipulate them is instrumental in many mathematical problem-solving scenarios. They are a common type of problem in algebra with practical applications across sciences and engineering.

Factoring

Factoring is a method used to rewrite expressions as the product of their factors. This is particularly useful when solving quadratic equations, where we aim to express them in the form of \((px + q)(rx + s) = 0\).

In this exercise, once we rearranged our quadratic equation \(x^2 - 20x + 36 = 0\), we factored it into \((x - 2)(x - 18) = 0\). Factoring enables us to easily derive potential solutions: in this case, \(x = 2\) and \(x = 18\).

Factoring is effective in simplifying expressions and solving equations by breaking them down into simpler components. It is a fundamental skill that helps students unlock solutions with clarity and confidence.

Verification of Solutions

After solving equations, it is vital to verify the solutions to ensure accuracy. Verification involves substituting solutions back into the original equation to check if they truly satisfy the equation.

In this problem, we verified two potential solutions. Substituting \(x = 2\) into the equation \(\sqrt[3]{\sqrt{32x}} = \sqrt[3]{x+6}\) proved valid since both sides equaled the same number. However, when substituting \(x = 18\), the sides did not match, indicating it was not a valid solution.

Verification ensures that our solutions are not just mathematically sound but also practically applicable to the given problem. It serves as a crucial checkpoint in the problem-solving process to confirm that we've reached the correct conclusion. This extra step provides confidence and completeness in our solution.