Problem 24 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Express the Equation with the Same Base

The first task is to express both sides of the equation with the same base. We know that \( 64 = 8^2 \). Since the right-hand side is \( \frac{1}{64} \), we can write it as \( 8^{-2} \). Thus, we rewrite the equation as \( 8^{-2x+1} = 8^{-2} \).

Step 2: Set the Exponents Equal

With both sides of the equation written with the same base, we can equate their exponents. Therefore, we have the equation \( -2x + 1 = -2 \).

Step 3: Solve for \( x \)

Now, solve the equation from the previous step. First, subtract 1 from both sides: \(-2x = -3\). Then divide both sides by \(-2\) to isolate \(x\): \(x = \frac{3}{2}\).

Key Concepts

Negative Exponents

When dealing with exponents, a negative exponent indicates that the number is the reciprocal of that number raised to the positive exponent. For instance, when you have an exponent like \[ a^{-n} \], this is equivalent to \[ \frac{1}{a^n} \]. This concept helps us rewrite fractions as exponential equations, which further simplifies solving problems.

In our exercise, the right side of the equation is given as \( \frac{1}{64} \). This can be rewritten using a negative exponent as \( 8^{-2} \), meaning that the original problem can be transformed by expressing both sides with the same base. This is crucial for tackling the solution since it lays the foundation for equalizing the exponents. Understanding negative exponents can help a lot in deconstructing complex fractional problems into more approachable equations.

Same Base Property

The same base property is a powerful tool when solving exponential equations. It states that if two exponents have the same base, you can set their exponents equal to each other. This property simplifies the equation significantly, reducing it to a simpler algebraic form.

In our exercise, both sides of the equation are transformed to the base of 8: \( 8^{-2x+1} \) and \( 8^{-2} \). Once they share the same base, solving the equation becomes more straightforward because you can ignore the base and only focus on the exponents:

• Set \( -2x+1 \) to equal \( -2 \).

Thus, we convert the problem from an exponential equation into a simple linear equation that’s much easier to solve.

Isolating Variables

Isolating variables is a fundamental technique in solving equations. The idea is to rearrange the equation in such a way that the variable you are solving for is alone on one side of the equation.

Once we have equalized the exponents by using the same base property, we set up the equation as \( -2x + 1 = -2 \). The next steps are straightforward:

• Subtract 1 from both sides to simplify: \(-2x = -3\).
• Divide both sides by -2 to isolate \( x \): \( x = \frac{3}{2} \).

Through these steps, we effectively solve for \( x \), completing the task of isolating the variable. Mastery of isolating variables will facilitate solving most algebraic equations, making it an invaluable skill in mathematics.