Problem 24 - Algebra for College Students

Question

Solve each of the equations. $$ \left(3^{x}\right)\left(3^{5 x}\right)=81 $$ \(\left\\{\frac{2}{3}\right\\}\)

Step-by-Step Solution

Answer

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

Step 1: Apply the Product of Powers Property

Recognize that you can use the property of exponents, which states that when you multiply like bases, you can add the exponents. Therefore, \(3^x \cdot 3^{5x} = 3^{x + 5x} = 3^{6x}\).

Step 2: Express 81 as a Power of 3

Remember that 81 is a power of 3, specifically \(81 = 3^4\). This allows us to write the original equation using the same base: \(3^{6x} = 3^4\).

Step 3: Set the Exponents Equal to Each Other

Since the bases in the equation \(3^{6x} = 3^4\) are the same, you can set the exponents equal to each other: \(6x = 4\).

Step 4: Solve for x

Divide both sides of the equation \(6x = 4\) by 6 to isolate \(x\): \[ x = \frac{4}{6} = \frac{2}{3} \].

Step 5: Verify the Solution

Substitute \(x = \frac{2}{3}\) back into the original equation to verify our solution:\[ \left(3^{\frac{2}{3}}\right) \left(3^{5(\frac{2}{3})}\right) = 3^{\frac{2}{3} + \frac{10}{3}} = 3^4 = 81 \]. The solution checks out.

Key Concepts

Understanding the Properties of Exponents

In mathematics, properties of exponents are fundamental tools that simplify expressions involving powers. These properties allow us to manipulate and solve equations efficiently. One crucial property is the **Product of Powers Property**. When you multiply two expressions that have the same base, you can add their exponents. For instance, if you have

• expressions like \( a^m \) and \( a^n \), their product is expressed as \( a^{m+n} \).

Applying this property simplifies the process of solving equations, as seen in the exercise where we combined \(3^x\) and \(3^{5x}\) to get \(3^{6x}\). By rewriting expressions using these properties, it's easier to identify relationships between the terms involved, leading us to the solution more directly.
Mastering these basic properties not only helps in solving problems faster but also enhances algebraic comprehension overall.

Step-by-Step Approach to Solving Equations

To solve exponential equations successfully, a structured approach is beneficial. The first step is often to express all terms with the same base, if possible. In the given problem, the terms are already in base 3. Observing that the number 81 is equivalent to \(3^4\) allows us to rewrite the equation consistently using this base.

• We have \(3^{6x} = 3^4\), therefore focusing entirely on the exponents simplifies the problem since the bases are now identical.
• Once the bases are equal, the next logical step is to equate the exponents themselves, which results in a simple algebraic equation.

From these observations, the task transitions from dealing with exponential forms to straightforward algebraic manipulation, enabling students to solve for \(x\). This methodical approach not only works for powers of 3 but is broadly applicable to similar problems involving different bases too.

Algebraic Techniques for Simplifying Equations

Using algebraic techniques is key in manipulating and solving exponential equations. In our exercise, we simplified the problem by transforming it into a linear equation using exponent rules. After aligning the bases to be equal (

• here through \(3^{6x} = 3^4\)),

we equated exponents, turning the problem into solving \(6x = 4\). This is a straightforward algebraic problem that can be solved using basic operations such as division.

• By dividing both sides by 6, we isolated \(x\), resulting in \(x = \frac{2}{3}\).

Practicing these techniques across different equations increases fluency in algebra, making solving more complex problems more manageable over time. A systematic approach to breaking down equations into simpler parts is invaluable, especially as the complexity of problems increases.

Exploring Powers and Roots in Equations

Powers, and roots, are integral in understanding and solving equations involving exponential terms. When dealing with powers like those in our exercise, recognizing how numbers can be rewritten as powers of similar bases is essential. For instance, expressing 81 as \(3^4\) was a crucial step in aligning the exponents, which drastically clarified the equation-solving process.

• This ability to convert numbers makes it easier to manipulate terms and solve equations using algebraic techniques.

Roots, on the other hand, come into play when solving equations leads to non-integer exponents, often requiring further simplification by taking roots. Although our specific exercise didn’t involve calculating roots beyond basic simplification, understanding how they are the inverse operations of powers can be crucial for more intricate equations. Mastery of rewriting numbers and understanding how powers and roots interact is vital for tackling exponential equations efficiently.