Problem 30 - Algebra 2 and Trigonometry

Question

Solve each equation and check. \(\left(\frac{1}{4}\right)^{x}=8^{1-x}\)

Step-by-Step Solution

Answer

The solution is \(x = 3\).

Step 1: Understand the Problem Statement

We are given the equation \(\left(\frac{1}{4}\right)^{x}=8^{1-x}\). Our goal is to find the value of \(x\) that satisfies this equation.

Step 2: Express Both Sides with the Same Base

The left side of the equation, \(\frac{1}{4}\), can be rewritten as \(4^{-1}\). The right side, \(8\), can be rewritten as \(2^3\), therefore \(8^{1-x}\) is \((2^3)^{1-x}\).

Step 3: Apply the Power of a Power Rule

Transform the expression by using the power of a power rule, \((a^m)^n = a^{m \times n}\). Thus, \(\left(4^{-1}\right)^x\) becomes \(4^{-x}\) and \((2^3)^{1-x}\) becomes \(2^{3(1-x)}\).

Step 4: Express in Terms of a Single Base

To express \(4\) as a base of 2, note that \(4 = 2^2\). So \(4^{-x} = (2^2)^{-x} = 2^{-2x}\). Therefore, the equation becomes \(2^{-2x} = 2^{3 - 3x}\).

Step 5: Solve for x by Equating Exponents

Since both sides of the equation have the same base \(2\), set their exponents equal: \(-2x = 3 - 3x\).

Step 6: Simplify the Exponential Equation

Add \(3x\) to both sides:\(-2x + 3x = 3\).

Step 7: Solve for x

Combine and simplify:\(x = 3\).

Step 8: Check the Solution

Substitute \(x = 3\) back into the original equation: \(\left(\frac{1}{4}\right)^{3} = 8^{1-3}\).Calculate: \(\frac{1}{4^3} = \frac{1}{64}\) and \(8^{-2} = \frac{1}{64}\).Since both sides are equal, \(x = 3\) is the correct solution.

Key Concepts

Power of a Power Rule

When you're working with exponential equations, the power of a power rule can be a very handy tool. This rule helps us simplify expressions where we have an exponent raised to another exponent. This situation usually looks like \((a^m)^n\). According to the power of a power rule, you multiply the exponents: \((a^m)^n = a^{m \times n}\). Let's see this in action with the equation from our exercise. Once both sides of the original equation are expressed with powers, applying this rule allows us to transform \((2^{3})^{1-x}\) into \(2^{3(1-x)}\).By simplifying in this way, complex exponential expressions become easier to manage, leading us ever closer to solving the equation.

Expressing in Terms of the Same Base

One powerful strategy for solving exponential equations is to rewrite them in terms of the same base. This means expressing different numbers using powers of a common base. In the exercise, we have the numbers \(4\) and \(8\). To express both sides of the equation in terms of base \(2\), we convert \(4\) into \(2^2\) and \(8\) into \(2^3\). Once expressed this way, the problem simplifies greatly.This essential technique lets us set the exponents equal to each other since the base is the same for both sides of the equation. Having the same base is crucial because the exponential function is one-to-one, meaning that if \(a^m = a^n\), then \(m = n\). This insight is fundamental in reducing the complexity of exponential equations.

Checking Solutions in Equations

It's important not to stop after solving an equation. Verifying the solution helps ensure that it's correct. After finding \(x = 3\), we substitute it back into the original equation to verify our work. Doing so, we find \((\frac{1}{4})^3 = 8^{1-3}\), which simplifies further to \(\frac{1}{64} = \frac{1}{8^2}\) or \(\frac{1}{64}\). Checking is a quick step that ensures no mistakes were made during calculations. By confirming both sides are equal with \(x = 3\), we solidify our solution. This crucial step does not only validate our work but also builds confidence in our understanding of the methods used.

Solving Exponential Equations

Solving exponential equations involves isolating the variable, usually an exponent, by manipulating the bases and applying algebraic principles. First, a given exponential equation might appear daunting, but strategic steps streamline the solution process.

• Express all terms using a common base, if possible. This reduces complexity.
• Use algebraic rules, like the power of a power, to simplify terms.
• Once terms share a common base, set the exponents equal.
• Solve the resulting linear equation for the variable.

These structured steps in our exercise brought about the solution with mathematical accuracy. Remember, different exponential equations might call for slightly different tactics, but the essence remains finding common ground among expressions. This methodical approach makes exponential equations less intimidating and more manageable.