Problem 49 - A Graphical Approach to College Algebra

Question

Solve each equation. $$\left(\frac{1}{4}\right)^{2-x}=2^{3 x+3}$$

Step-by-Step Solution

Answer

The solution is \(x = -7\).

Step 1: Rewrite the base

Rewrite \( \left(\frac{1}{4}\right)^{2-x} \) as \((4^{-1})^{2-x}\), which simplifies to \(4^{-(2-x)}\). Since \(4 = 2^2\), this can be rewritten as \((2^2)^{-(2-x)} \).

Step 2: Apply the Power of a Power Property

Using the power of a power property, \((a^m)^n = a^{m \cdot n}\), rewrite \((2^2)^{-(2-x)}\) as \(2^{2 \cdot -(2-x)}\), which becomes \(2^{-2(2-x)}\).

Step 3: Simplify the Exponent

Simplify the exponent: \(-2(2-x) = -4 + 2x\). So, the expression becomes \(2^{2x - 4}\).

Step 4: Set the Exponents Equal

Since the bases are the same, set the exponents equal: \[2x - 4 = 3x + 3\].

Step 5: Solve for x

Subtract \(2x\) from both sides to get: \(-4 = x + 3\). Now, subtract \(3\) from both sides to isolate \(x\):\[-4 - 3 = x\]. Thus, \(x = -7\).

Key Concepts

Power of a Power Property

When dealing with exponential equations, one useful property we often rely on is the power of a power property. This property allows us to simplify complex exponents by turning them into more manageable forms. The power of a power property is expressed as \[(a^m)^n = a^{m \cdot n}\].This means that when you have an exponent raised to another exponent, you can multiply the exponents.

For example, if you have \(((2^2)^{-(2-x)})\), it can be rewritten using this property as \(2^{2 \cdot -(2-x)}\). By simplifying further, you can get a simpler exponent \(2^{-2(2-x)} = 2^{2x - 4}\).

This property is incredibly important, as it helps you set up the equation in a way that allows for easier solving. Make sure to understand and apply this property effectively to simplify the steps in solving exponential equations.

Base Rewriting

Base rewriting is a critical step in solving exponential equations as it aligns the bases so we can compare exponents directly. When you have different bases like \(\left(\frac{1}{4}\right)\) and \(2\) in an equation, rewriting them to share a common base can simplify the process.

First, rewrite the fraction, \(\left(\frac{1}{4}\right)\) as \(4^{-1}\), because \(\frac{1}{4}\) is the same as \(4\) raised to the power of \(-1\). Then, recognize that \(4\) can be written as \(2^2\), which aligns it with the base of \(2\) on the other side of the equation.

So, you convert \(\left(\frac{1}{4}\right)^{2-x}\) into \((2^2)^{-(2-x)}\). This transformation using common bases is essential as it allows us to apply the power properties and solve the equation more smoothly.

Solving for x

Once the bases are rewritten and simplified, your next step is solving for the unknown variable, often \(x\) in the equation. When both sides of the equation have the same base, for example, \(2\), you can set their exponents equal to each other, as \[2x - 4 = 3x + 3\].

This equation now becomes a linear one, where moving terms around to isolate \(x\) is straightforward:

• First, subtract \(2x\) from both sides to simplify: \(-4 = x + 3\).
• Next, to isolate \(x\), subtract \(3\) from both sides: \(-4 - 3 = x\).

This leaves you with \(x = -7\).

Days where \(x\) seems invincible are behind you when you break it down to simple arithmetic. By keeping the equation linear and straightforward, you can quickly find your solution.