Problem 72 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

The real solutions are \( x = 0, x = 1, \) and \( x = -1 \).

Step 1: Understand the Nested Radicals

The given equation is \( \sqrt[3]{\sqrt[3]{x}} = x \). It involves two cube roots, a nested radical situation. Our goal is to simplify this to find all possible real solutions for \( x \).

Step 2: Convert Nested Radicals

Let \( y = \sqrt[3]{x} \). Then the equation becomes \( \sqrt[3]{y} = x \). Since \( y = \sqrt[3]{x} \), it follows that \( y^3 = x \). From \( \sqrt[3]{y} = x \), we also have \( y = x^3 \). Thus, we have two equations: \( y^3 = x \) and \( y = x^3 \).

Step 3: Solve for y

From the equations \( y^3 = x \) and \( y = x^3 \), substitute \( y = x^3 \) into the first equation to get \( (x^3)^3 = x \). Simplifying gives \( x^9 = x \).

Step 4: Solve the Simplified Equation

The equation \( x^9 = x \) can be rearranged to \( x^9 - x = 0 \). Factoring gives \( x(x^8 - 1) = 0 \). This results in two cases: \( x = 0 \) and \( x^8 - 1 = 0 \).

Step 5: Solve Each Case for x

For \( x = 0 \), we directly get one solution: \( x = 0 \). For the equation \( x^8 - 1 = 0 \), note that \( x^8 = 1 \), so \( x = 1 \) or \( x = -1 \) are solutions, as \( x^8 = (1)^8 = 1 \) and \( x^8 = (-1)^8 = 1 \).

Step 6: Review and Confirm Solutions

The possible real solutions are \( x = 0, x = 1, \) and \( x = -1 \). To confirm, check each solution within the original equation. Each satisfies the equation, confirming their validity.

Step 7: Graphical Verification

Graph the functions \( f(x) = \sqrt[3]{\sqrt[3]{x}} \) and \( g(x) = x \). The points where these graphs intersect correspond to the solutions found: \( x = 0, x = 1, \) and \( x = -1 \).

Key Concepts

Solving Equations

When faced with equations involving nested radicals, the primary objective is to simplify them into a more manageable form. Nested radicals can seem daunting at first, but with systematic steps, they become solvable.

The key is to break down the nested structure. For instance, in the problem \( \sqrt[3]{\sqrt[3]{x}} = x \), there's a radical nested within another. A good starting strategy is to introduce a substitution that tidies the equation. If we let \( y = \sqrt[3]{x} \), the equation transforms to simpler terms, \( \sqrt[3]{y} = x \).

This step exposes the internal structure of the equation, allowing you to express each part in terms of variables and constants. Next, relate these expressions back to the original terms. As you substitute back, look for opportunities to equate powers on both sides, a central tactic in solving such radicals.

• Recognize substitution opportunities.
• Express nested terms in simpler forms.
• Solve for your substituted variables.

Cube Roots

A cube root is a number that, when multiplied by itself three times, gives the original number. Understanding cube roots is crucial when dealing with equations like \( \sqrt[3]{x} \).

Since we are dealing with cube roots, it's essential to remember that both negative and positive numbers have real cube roots. This property is handy when finding solutions to equations, as you might come across both positive and negative solutions despite dealing with even-powered equations.

• Cube roots apply to both positive and negative numbers.
• The cube of the cube root returns the original number.

In practice, when simplifying expressions involving cube roots, always focus on expressing terms such that each one matches the power necessary to neutralize the cube root. This consideration makes solving nested equations straightforward as you continue to unearth the structure step-by-step.

Algebraic Simplification

Algebraic simplification is the process of reducing an expression or equation to its simplest form. When dealing with equations like \( x^9 = x \), it is often useful to bring all terms to one side to identify common factors.

In this problem, the simplification led to the equation \( x^9 - x = 0 \), which could be factored into \( x(x^8 - 1) = 0 \). Factoring is a crucial step as it assists in breaking down complex equations into simpler, easily solvable parts.

Looking at the factors, \( x = 0 \) becomes one solution straightforwardly, while \( x^8 - 1 = 0 \) simplifies further to help find additional solutions. Recognize that \( x^8 = 1 \) implies that \( x = 1 \) or \( x = -1 \) since these numbers, when raised to the eighth power, give \( 1 \).

• Bring terms to one side of the equation.
• Factor the equation for simpler parts.
• Identify and solve resulting simpler equations.