Problem 5 - Intermediate Algebra

Question

For Problems \(1-34\), solve each equation. $$ \left(\frac{1}{4}\right)^{x}=\frac{1}{256} $$

Step-by-Step Solution

Answer

x = 4

Step 1: Understand the Equation

The equation given is \( \left(\frac{1}{4}\right)^{x} = \frac{1}{256} \). We need to solve for \( x \), meaning we must find the value of \( x \) that makes this equality true.

Step 2: Express Terms with the Same Base

Notice that both sides can be expressed in terms of base 4. We recognize that \( \frac{1}{4} = 4^{-1} \) and \( 256 = 4^{4} \). Therefore, \( \frac{1}{256} = (4^4)^{-1} = 4^{-4} \). So our equation becomes \( \left(4^{-1}\right)^{x} = 4^{-4} \).

Step 3: Apply the Power of a Power Property

Using the property of exponents, \( (a^m)^n = a^{mn} \), we can write \( \left(4^{-1}\right)^{x} = 4^{-1 \cdot x} = 4^{-x} \). Our equation now is \( 4^{-x} = 4^{-4} \).

Step 4: Equate the Exponents

Since the bases are equal, we can set the exponents equal to each other: \(-x = -4\).

Step 5: Solve for x

Solving the equation \(-x = -4\) gives us \( x = 4 \).

Key Concepts

Power of a Power Property

When dealing with exponential expressions, one useful tool is the power of a power property. This property allows us to simplify expressions of the form \((a^m)^n\). According to this rule, we can rewrite \((a^m)^n\) as \(a^{m \cdot n}\).

In our problem, we encountered \(\left(4^{-1}\right)^{x}\). Utilizing the power of a power property, we rewrote this as \(4^{-1 \cdot x} = 4^{-x}\).

This transformation makes it easier to handle equations by straightening out the exponential expressions into a simpler, more manageable form. When an expression is simplified like this, comparing it with other exponential terms becomes more straightforward.

Express Terms with the Same Base

A crucial step in solving exponential equations is expressing terms with the same base. This tactic helps in equating different expressions. It involves writing both sides of an equation as powers of the same base which simplifies comparisons and calculations.

In the exercise \(\left(\frac{1}{4}\right)^{x} = \frac{1}{256}\), we start by transforming the terms so that both are powers of 4.

• Recognize \(\frac{1}{4}\) as \(4^{-1}\), because it's the reciprocal of \(4\).
• Express \(\frac{1}{256}\) as a power of 4. Since \(256 = 4^4\), its reciprocal is \(4^{-4}\).

By rewriting both sides in this manner, the expression becomes \(\left(4^{-1}\right)^{x} = 4^{-4}\). Now, with a common base of 4 on both sides, we can move to the next step.

Solving for x

Once both sides of an exponential equation share the same base, we can solve for the variable \(x\) by setting the exponents equal.

In our problem, after equating \(4^{-x} = 4^{-4}\), we realize that because the bases are identical, the exponents must be equal:

• This gives us the equation \(-x = -4\).

Solving this linear equation is straightforward. Multiply both sides by (-1) to achieve \(x = 4\), which satisfies the original equation.

Solving for \(x\) involves identifying equivalent expressions and manipulating them algebraically until you isolate \(x\). Through consistent practice, it becomes simpler to tackle similar problems, fostering a stronger understanding of manipulating exponential equations.