Problem 5 - College Algebra with Corequisite Support

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 value of \(x\) is \(-\frac{1}{3}\).

Step 1: Express Each Number as Powers of a Common Base

First, express all the numbers as powers of a common base. Both 64 and 16 can be expressed as powers of 4. We know that:\[64 = 4^3\] and\[16 = 4^2\]Thus, rewrite the equation using these expressions:\[4^3 \cdot 4^{3x} = 4^2\]

Step 2: Combine the Exponents

Since the bases are the same, we can combine the exponents on the left side of the equation. According to the properties of exponents:\[4^{3+3x} = 4^2\]This simplifies to:\[4^{3x+3} = 4^2\]

Step 3: Set Exponents Equal to Each Other

When the bases are the same and the expression is equal, you can set the exponents equal to each other:\[3x + 3 = 2\]

Step 4: Solve for x

Isolate the variable \(x\) in the equation:Subtract 3 from each side:\[3x = 2 - 3\] \[3x = -1\]Then, divide by 3:\[x = -\frac{1}{3}\]

Key Concepts

Like Bases

When solving exponential equations, one effective way to simplify them is by using like bases. Like bases mean that each term in the equation is rewritten to have the same base, which can then be easily manipulated. For example, in the original exercise with the equation \(64 \cdot 4^{3x} = 16\), both 64 and 16 can be rewritten using a base of 4. Why choose 4? Well, if you break 64 and 16 down into factors of 4, you find:

• \(64 = 4^3\)
• \(16 = 4^2\)

By rewriting the equation using the same base, you set the stage to simplify and solve it more effectively. In other words, having like bases helps us apply the properties of exponents to ultimately solve for the unknown variable. This is a crucial first step when dealing with exponential equations as it gives us a straightforward path to manipulate the equation.

Properties of Exponents

The properties of exponents are incredibly beneficial when working with equations involving powers. These properties allow us to manipulate expressions in simpler ways. One key property is when multiplying like bases, you add their exponents. For example, in our equation \(4^3 \cdot 4^{3x} = 4^2\), since the bases are the same, the exponents can be combined:

• \(4^{3+3x} = 4^{2}\)

This crucial step greatly simplifies the original equation. It's important to master this and other properties:

• \((a^m)^n = a^{mn}\)
• \(a^m \cdot a^n = a^{m+n}\)
• \(a^m / a^n = a^{m-n}\)

Learning how to apply these properties provides immense help in various algebraic situations, particularly in problems involving exponential functions and equations.

Solving for Variables

Once you've successfully used like bases and properties of exponents, the next step is to solve for the unknown variable. In the given problem, after simplifying the equation to \(4^{3x+3} = 4^{2}\), you can equate the exponents since the bases are the same:

• \(3x + 3 = 2\)

By setting the exponents equal to each other, you turn the problem into a simple linear equation, making it much more manageable.Here's the process to solve for \(x\): Subtract 3 from both sides:

• \(3x = 2 - 3\)
• \(3x = -1\)

Finally, divide by 3:

• \(x = -\frac{1}{3}\)

Solving for the variable involves simplifying the equation step by step until \(x\) stands alone. This process applies not only to exponential equations but to any equation where isolating a variable is required. Mastering this technique is essential for solving equations efficiently.