Problem 53 - A Graphical Approach to College Algebra

Question

Solve each equation. $$(\sqrt{2})^{-2 x}=\left(\frac{1}{2}\right)^{2 x+3}$$

Step-by-Step Solution

Answer

The solution is \( x = -3 \).

Step 1: Understand the Bases

First, recognize that the square root of 2, \(\sqrt{2}\), can be expressed as \(2^{1/2}\). Meanwhile, \( \frac{1}{2} \) can be expressed as \(2^{-1}\). This understanding will help us in finding common bases for the exponents.

Step 2: Rewrite Both Sides with a Common Base

Express \((\sqrt{2})^{-2x}\) as \((2^{1/2})^{-2x}\), which simplifies to \((2^{-x})\). Likewise, express \(\left( \frac{1}{2} \right)^{2x+3}\) as \(2^{-(2x+3)}\). This gives us the equation \(2^{-x} = 2^{-(2x+3)}\).

Step 3: Equate the Exponents

Since we have the same base on both sides of the equation, we can equate the exponents: \(-x = -(2x + 3)\). This simplifies to \(-x = -2x - 3\).

Step 4: Solve for x

To isolate \(x\), add \(2x\) to both sides of the equation: \(-x + 2x = -2x + 2x - 3\)which simplifies to \(x = -3\).

Key Concepts

Solving Equations

Equations are mathematical statements that assert the equality of two expressions. In this case, we're dealing with exponential equations, which involve expressions with variables in the exponent. The goal is to find the value of the variable that makes the equation true.

Here’s a simplified approach to solving these equations:

• Identify the expressions on both sides of the equation.
• Try to express both sides using the same base. This makes it easier to compare the exponents directly.
• Once both sides have a common base, set their exponents equal to each other to solve for the unknown variable.
• Complete the algebraic manipulations to find the variable's value.

In the given problem, we transform the bases to a common base, making it easier to solve.

Exponent Rules

Understanding exponent rules is crucial when working with exponential equations. These rules help us simplify and manipulate expressions efficiently. Here are some key rules:

• Product of Powers: If the bases are the same, add the exponents: \(a^m \times a^n = a^{m+n}\).
• Power of a Power: Multiply the exponents when raising a power to another power: \((a^m)^n = a^{mn}\).
• Power of a Product: Distribute the exponent to each factor: \((ab)^n = a^n b^n\).
• Negative Exponent: Represents reciprocal: \(a^{-n} = \frac{1}{a^n}\).

Applying these rules allowed us to rewrite \((\sqrt{2})^{-2x}\) as \(2^{-x}\) and \(\left(\frac{1}{2}\right)^{2x+3}\) as \(2^{-(2x+3)}\), making it possible to directly set their exponents equal.

Algebraic Manipulation

Algebraic manipulation involves performing operations to simplify expressions and solve equations. Let's break down the process using our example:

1. **Rewriting with Common Bases:** We changed \((\sqrt{2})^{-2x}\) to \(2^{-x}\) and \(\left(\frac{1}{2}\right)^{2x+3}\) to \(2^{-(2x+3)}\). This was possible because we used exponent rules.
2. **Equate the Exponents:** Once both sides had the same base, we equated the exponents: \(-x = -(2x + 3)\).
3. **Solving for x:** We needed to isolate \(x\). By adding \(2x\) to each side, we simplified the equation to \(x = -3\).

Through algebraic manipulation, we systematically solve for the value of the variable, ensuring the equation remains balanced throughout the process.