Problem 18 - Intermediate Algebra

Question

For Problems \(1-34\), solve each equation. $$ \left(\frac{1}{3}\right)^{5 x}=243 $$

Step-by-Step Solution

Answer

The solution is x = -1.

Step 1: Rewrite the Equation with Same Base

First, recognize that 243 can be written as a power of 3. We know that \(243 = 3^5\). Now rewrite the equation: \( \left(\frac{1}{3}\right)^{5x} = 3^5 \).

Step 2: Express as Negative Exponent

Notice that \(\frac{1}{3}\) can be rewritten as \(3^{-1}\). Substitute this into the equation to get: \((3^{-1})^{5x} = 3^5\).

Step 3: Apply the Power Rule

Using the power rule, \((a^{m})^{n} = a^{mn}\), we simplify the left side: \((3^{-1})^{5x} = 3^{-5x}\). Thus, the equation becomes \(3^{-5x} = 3^5\).

Step 4: Set the Exponents Equal

Since the bases are the same, set the exponents equal to each other: \(-5x = 5\).

Step 5: Solve for x

Solve the equation \(-5x = 5\) for \(x\) by dividing both sides by \(-5\): \(x = -1\).

Key Concepts

Understanding Negative Exponents

Negative exponents can seem daunting at first, but they're just a different way to express fractions. The principle behind negative exponents is a simple one: when you have a base raised to a negative exponent, it is the same as the reciprocal of the base raised to the positive equivalent of that exponent. For example,

• \( a^{-n} = \frac{1}{a^n} \), where \(a\) is any non-zero number and \(n\) is a positive integer.

So, when you see something like \( \left(\frac{1}{3}\right)^{5x} \), you can rewrite the base \(\frac{1}{3}\) as \(3^{-1}\). By understanding this transformation, you can turn complex looking fractions into more manageable forms using negative exponents.

Applying the Power Rule

The power rule is an exciting tool in the world of exponentiation. It helps in simplifying expressions where an exponent is raised to another power. The power rule states that:

• \((a^{m})^{n} = a^{mn}\)

This means when you have a base raised to a power, and that entire expression is raised to another power, you multiply the exponents together. For instance, in expression \((3^{-1})^{5x}\), applying the power rule results in \(3^{-5x}\).
Why is this useful? It simplifies solving exponential equations by allowing us to deal with a single base power. Learning how to apply this rule effectively can make solving even the more complex problems seem straightforward.

Rewriting as Same Base

Rewriting each side of an equation with the same base can make solving exponential equations much simpler. When the bases are the same, you can compare and solve just the exponents. This is often the key to solving these problems efficiently. In our case, 243 can be rewritten as a power of 3, specifically:

• \(243 = 3^5\)

By rewriting \(\left(\frac{1}{3}\right)^{5x}\) using the base 3 as \(3^{-5x}\), the original exercise becomes manageable, setting us up to compare exponents directly. Remembering that the same base allows you to equate the exponents is crucial to taking control of the problem.

Solving the Exponential Equation

Once you've reduced the problem to having the same base on both sides, solving exponential equations becomes quite straightforward. With both sides rewritten as \(3^{-5x} = 3^{5}\), the bases cancel out, leaving the exponents:

• \(-5x = 5\)

This equation is now a simple linear equation to solve. You continue by isolating \(x\). Divide both sides by -5 to find:

• \(x = -1\)

When solving exponential equations, the bulk of the work lies in setting the equation to an equal base. Once you achieve that, the solution becomes a matter of basic algebra. It's all about simplifying and isolating the variable.