Problem 24 - Intermediate Algebra

Question

Solve each equation. $$ 8^{-2 x+1}=\frac{1}{64} $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{3}{2} \).

Step 1: Convert the right side to base 8

Since we have the base as 8 on the left side, start by writing the number 64 in base 8. We know \( 8^2 = 64 \), so we can rewrite \( \frac{1}{64} \) as \( 8^{-2} \). Thus, the equation becomes \( 8^{-2x+1} = 8^{-2} \).

Step 2: Set the exponents equal to each other

Now that we have the same base on both sides of the equation, set the exponents equal to each other: \( -2x + 1 = -2 \).

Step 3: Solve for \( x \)

Solve the equation \( -2x + 1 = -2 \) for \( x \).First, subtract 1 from both sides:\[ -2x + 1 - 1 = -2 - 1 \]This simplifies to:\[ -2x = -3 \]Next, divide by -2:\[ x = \frac{-3}{-2} \]This simplifies to:\[ x = \frac{3}{2} \]

Key Concepts

Properties of Exponents

When dealing with exponential equations, understanding the properties of exponents is essential. These properties allow us to manipulate and solve equations that feature exponential expressions. Here are some key properties:

• Product of Powers: When multiplying like bases, \( a^m \cdot a^n = a^{m+n} \), you add the exponents.
• Power of a Power: Raising a power to another power, \((a^m)^n = a^{m\cdot n}\), requires that you multiply the exponents.
• Power of a Product: This states that \((ab)^m = a^m \cdot b^m\), which applies the exponent to each factor in the product.
• Negative Exponent: \(a^{-m} = \frac{1}{a^m}\), essentially flips the base into a reciprocal form.
• Zero Exponent: Any nonzero base raised to the zero power equals one, \(a^0 = 1\).

Recognizing these properties can help simplify and solve equations like the one in the exercise, where the base of the exponents was identical on both sides.

Base Conversion

Base conversion is a technique used to change numbers from one base to another. This is particularly useful when solving exponential equations where both sides must share the same base. Let's consider some basics:
1. **Identifying the Base**: To perform base conversion, you first need to determine if there's an existing base commonality.2. **Applying the Appropriate Power**: Once you know the base, express numbers in terms of powers of that base.
In the exercise, the original problem required converting \( \frac{1}{64} \) to base 8. Recognizing that \( 64 = 8^2 \) is the key because it allows you to rewrite it as \( 8^{-2} \). This step, where both sides now share the same base, sets the stage for solving the equation.
Base conversion is crucial in creating an equation that is simpler to solve, as having identical bases allows us to directly equate the exponents.

Algebraic Equations

Algebraic equations involve variables and constants, structured around operations like addition, subtraction, multiplication, and division. Solving these equations often requires a methodical approach to isolating the variable. Here is a simple strategy:

• **Equate the Exponents**: Once identical bases are established, equate the exponents from both sides.
• **Isolate the Variable**: Rearrange the equation to get the variable alone on one side. This typically involves performing the opposite operation. For example, if the variable is subtracted, add to both sides.
• **Solve for the Variable**: Use basic algebraic operations, such as addition, subtraction, multiplication, or division, to find the solution.

In our exercise, after converting both sides to the same base, we isolated the variable by equating \(-2x + 1 = -2\). Solving this for \(x\) involved subtracting 1 from both sides and then dividing by -2, giving us \(x = \frac{3}{2}\). Understanding these steps will enable you to tackle similar algebraic equations confidently.