Problem 5 - College Algebra

Question

For the following exercises, use like bases to solve the exponential equation. $$ 64 \cdot 4^{3 x}=16 $$

Step-by-Step Solution

Answer

The solution is \(x = -\frac{1}{3}\).

Step 1: Express Each Number as a Power of 2

First, express 64 and 16 as powers of 2. We have: \[64 = 2^6\] and \[16 = 2^4\]. These expressions allow us to rewrite the equation using a single base.

Step 2: Rewrite the Equation with Base 2

Rewrite the original equation \(64 \cdot 4^{3x} = 16\) using powers of 2. Note that \(4 = 2^2\), so \(4^{3x} = (2^2)^{3x} = 2^{6x}\).The equation becomes:\[2^6 \cdot 2^{6x} = 2^4\].

Step 3: Apply the Product of Powers Property

Using the property \(a^m \cdot a^n = a^{m+n}\), combine the powers of 2 on the left side:\[2^{6+6x} = 2^4\].

Step 4: Set the Exponents Equal to Each Other

Since the bases are the same, we can set the exponents equal:\[6 + 6x = 4\].

Step 5: Solve for x

Subtract 6 from both sides to isolate the term with \(x\):\[6x = 4 - 6\],which simplifies to:\[6x = -2\].Now, divide both sides by 6 to solve for \(x\):\[x = \frac{-2}{6} = -\frac{1}{3}\].

Key Concepts

Like Bases

In the realm of exponential equations, understanding the concept of like bases is crucial to simplifying and solving these equations effectively. A like base refers to having the same number as a base for the exponential terms in an equation. Let's break this down through an example:

Consider the equation \(64 \cdot 4^{3x}=16\). To solve this equation using like bases, we first need to express all terms using the same base. In this case, we convert 64, 4, and 16 into powers of 2.

• 64 can be rewritten as \(2^6\)
• 4 can be rewritten as \(2^2\)
• 16 can be rewritten as \(2^4\)

This step allows us to rewrite the equation, making all bases consistent, specifically here as \(2\). This will greatly simplify the process of solving the equation as we can now focus on manipulating the exponents, knowing they all relate to a common base.

Powers of 2

The powers of 2 are an especially important tool in solving exponential equations. Recognizing and expressing numbers as powers of 2 can streamline the solution process. A power of 2 is any integer that can be expressed as \(2^n\), where \(n\) is a non-negative integer.

In our example, we used powers of 2 to tackle the equation \(64 \cdot 4^{3x}=16\). By expressing:

• 64 as \(2^6\)
• 4 as \(2^2\), leading to \(4^{3x} = (2^2)^{3x} = 2^{6x}\)
• 16 as \(2^4\)

We converted all parts of the equation to a consistent base, \(2\). This sets up for simpler manipulation, allowing us to focus on comparing exponent values rather than dealing with varied bases. Understanding this concept hinges on recognizing common powers of 2, helping us simplify many exponential scenarios.

Product of Powers Property

The "Product of Powers Property" is an essential property in algebra that facilitates the simplification of equations involving exponents. It states that when you multiply powers with the same base, you can add their exponents. Formally, this property is expressed as \(a^m \cdot a^n = a^{m+n}\).

In our example, when we rewrite the equation as \(2^6 \cdot 2^{6x} = 2^4\), this property allows us to condense the left-hand side into a single exponent expression \(2^{6 + 6x}\).

Here's how it looks in practice:

• Apply the property: \(2^6 \cdot 2^{6x} = 2^{6+6x}\)
• We then have \(2^{6+6x} = 2^4\)

Once you have a single base on both sides of the equation, you can equate the exponents directly since the bases match. This property is invaluable for simplifying equations swiftly and accurately, especially when dealing with exponential terms.