Question

Solve each equation. Express irrational solutions in exact form. $$ (\sqrt[3]{2})^{2-x}=2^{x^{2}} $$

Short answer

The solutions are \( x = \frac{2}{3} \) and \( x = -1 \).

Step-by-step solution

Identify the base and exponents

Rewrite the equation with the same base. Note that \( \sqrt[3]{2} = 2^{\frac{1}{3}} \ \). So the equation becomes: \[ (2^{\frac{1}{3}})^{2-x} = 2^{x^{2}} \]

Simplify the left-hand side

Apply the power rule \( (a^m)^n = a^{m \cdot n} \) to the left-hand side of the equation: \[ 2^{\frac{1}{3} \cdot (2-x)} = 2^{x^{2}} \]

Set the exponents equal

Since the bases are the same, set the exponents equal to each other: \[ \frac{1}{3} (2-x) = x^{2} \]

Clear the fraction

Multiply every term by 3 to clear the fraction: \[ 2 - x = 3x^{2} \]

Rearrange into a standard quadratic equation

Rearrange the equation into the standard form \( ax^2 + bx + c = 0 \): \[ 3x^{2} + x - 2 = 0 \]

Solve the quadratic equation

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to solve for x. Here, \( a = 3 \), \( b = 1 \), and \( c = -2 \). \[ x = \frac{-1 \pm \sqrt{1 + 24}}{6} = \frac{-1 \pm \sqrt{25}}{6} = \frac{-1 \pm 5}{6} \]

Find the solutions

Calculate the two possible values for x: \[ x = \frac{-1 + 5}{6} = \frac{4}{6} = \frac{2}{3} \quad \text{and} \quad x = \frac{-1 - 5}{6} = \frac{-6}{6} = -1 \]

Key concepts

Understanding Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. In algebra, those symbols (often represented by letters) are used to represent numbers in equations and formulas. The basic operations in algebra include addition, subtraction, multiplication, and division.

When solving equations, our main goal is to find the value of the variable that makes the equation true. For example, in the algebraic equation \( 2x + 3 = 7 \), we want to find the value of \( x \). To do so, we perform operations that will isolate \( x \) on one side of the equation.

Key algebra concepts include:

• Variables and Constants: Letters like \( x \) or \( y \) are variables, and specific values like 2 or 5 are constants.
• Expressions: Combinations of variables and constants, like \( 2x + 3 \).
• Equations: Statements that two expressions are equal, like \( 2x + 3 = 7 \).
• Operations: Addition, subtraction, multiplication, and division used to manipulate equations and expressions.

Understanding these basic concepts is crucial because they form the foundation of more complex topics, such as solving quadratic equations and working with exponents.

Solving Quadratic Equations

A quadratic equation is a second-order polynomial equation in a single variable \( x \) with the general form: \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. Solving a quadratic equation means finding the values of \( x \) that satisfy the equation. There are several methods to solve quadratics, including:

• Factoring: Expressing the quadratic as a product of its linear factors, if possible.
• Completing the square: Rewriting the quadratic equation so that one side is a perfect square trinomial.
• Quadratic Formula: Using the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the solutions directly.

For example, in our problem:

\[ 3x^2 + x - 2 = 0 \] We use the quadratic formula with \( a = 3 \), \( b = 1 \), and \( c = -2 \). Applying the formula, we get: \[ x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 3 \cdot (-2)}}{2 \cdot 3} \] Simplifying further: \[ x = \frac{-1 \pm \sqrt{25}}{6} \], which gives us \( x = \frac{2}{3} \) and \( x = -1 \). Quadratic equations often have two solutions, as shown here.

Working with Exponents

Exponents are a way to express repeated multiplication of the same number. For example, \(2^3\) means \(2 \times 2 \times 2 = 8\). In general, \(a^n\) means \(a\) multiplied by itself \(n\) times.

Key properties and rules of exponents that are essential include:

• Product Rule: \(a^m \cdot a^n = a^{m+n}\)
• Power Rule: \((a^m)^n = a^{m \cdot n}\)
• Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}\)
• Zero Exponent Rule: \(a^0 = 1\) for any \(a e 0\)
• Negative Exponent Rule: \(a^{-n} = \frac{1}{a^n}\)

Understanding these rules helps simplify expressions involving exponents. In our problem, we are given: \[(\sqrt[3]{2})^{2-x} = 2^{x^2}\] Recognizing that \(\sqrt[3]{2} = 2^{1/3}\), we can rewrite the equation: \[(2^{1/3})^{2-x} = 2^{x^2}\] Using the Power Rule, the left-hand side becomes \(2^{(1/3) \cdot (2-x)}\), leading us to set the exponents equal since the bases are the same: \[\frac{1}{3}(2-x) = x^2\]. Clearing the fraction by multiplying by 3, we work toward solving the quadratic equation \(3x^2 + x - 2 = 0\). Fully understanding exponents is crucial for solving various algebraic problems, especially those involving exponential equations.