Problem 32 - Intermediate Algebra

Question

For Problems \(1-34\), solve each equation. $$ \left(3^{x}\right)\left(3^{5 x}\right)=81 $$

Step-by-Step Solution

Answer

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

Step 1: Using properties of exponents

The equation is \( (3^x)(3^{5x}) = 81 \). We can use the property of exponents which states that \( a^m \times a^n = a^{m+n} \). So, \( 3^x \times 3^{5x} = 3^{x+5x} = 3^{6x} \).

Step 2: Express 81 as a power of 3

Now, we need to express 81 as a power of 3. We know that \( 81 = 3^4 \). Therefore, the equation becomes \( 3^{6x} = 3^4 \).

Step 3: Equate exponents

Since the bases are the same (both are 3), we can equate the exponents: \( 6x = 4 \).

Step 4: Solve for x

To solve for \( x \), divide both sides of the equation \( 6x = 4 \) by 6: \[ x = \frac{4}{6} \]. Simplify the fraction: \[ x = \frac{2}{3} \].

Key Concepts

Solving Equations

Solving equations is a fundamental skill in mathematics that involves finding the value of unknown variables. In our exercise, the goal is to determine the value of \( x \) that satisfies the given equation \( (3^x)(3^{5x}) = 81 \).
To do this, we apply different mathematical strategies that help us isolate and compute the unknown variable. One common method is to equate exponents when the bases are the same, which greatly simplifies the process. Once we have a simplified equation with equal exponents, solving for the unknown is straightforward. The key is to manipulate the equation step by step, gradually isolating the variable on one side of the equation, ensuring each step maintains equality.
This often involves dividing, multiplying, adding, or subtracting both sides by the same number, as demonstrated when solving for \( x = \frac{2}{3} \) in our example.

Properties of Exponents

Understanding the properties of exponents is crucial when dealing with equations that involve power expressions. Exponents have several rules and properties that make calculations easier.
In our problem, we used one such property: \( a^m \times a^n = a^{m+n} \). This rule is very useful because it allows us to combine expressions that share the same base, simplifying a more complicated expression into a single expression with one exponent.
Another important property to understand is that when bases are equal, and they are set equal in an equation, their exponents must also be equal. This allows us to equate the exponents directly, as seen with \( 3^{6x} = 3^4 \) from the original equation. Recognizing and applying these properties effectively can simplify seemingly complex problems.

Simplifying Expressions

Simplifying expressions is about making them as concise and straightforward as possible without changing their value. In our equation, simplification involved several important steps.
First, we simplified the multiplication of two exponential terms using the exponents' property, converting \((3^x)(3^{5x})\) to \(3^{6x}\). This turned a multiplication operation into an addition operation in the exponent.
Next, expressing numbers in terms of powers of a common base was key. The number 81 was successfully written as \(3^4\). This allowed for an easy comparison and reduction of the equation. Finally, simplifying the fraction \(x = \frac{4}{6}\) to \(x = \frac{2}{3}\) reduced the expression to its simplest form.
Through careful simplification, complex expressions become easier to work with, making it much easier to find a solution to a problem.