Problem 47 - Algebra for College Students

Question

Explain how you would solve the equation $$ \left(2^{x+1}\right)\left(8^{2 x-3}\right)=64 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 2\).

Step 1: Express 8 and 64 as Powers of 2

First, we need to express all numbers in the equation in terms of powers of 2 for consistency. We know that \(8 = 2^3\) and \(64 = 2^6\). Thus, we can rewrite the equation as \((2^{x+1})(2^{3(2x-3)}) = 2^6\).

Step 2: Simplify the Left Side of the Equation

Using the property of exponents \(a^m\cdot a^n = a^{m+n}\), simplify the left side of the equation. The first term \(2^{x+1}\) is \(2^x\cdot 2^1\). The second term \(2^{3(2x-3)}\) is \(2^{6x-9}\). Thus, we have \(2^{x+1 + 6x-9} = 2^6\).

Step 3: Combine Exponents

Combine the exponents on the left side: \((x+1) + (6x-9)\) simplifies to \(7x - 8\). Therefore, the equation becomes \(2^{7x-8} = 2^6\).

Step 4: Set Exponents Equal

Since the bases are the same, we can set the exponents equal to each other: \(7x - 8 = 6\). Solve this equation for \(x\).

Step 5: Solve for x

Add 8 to both sides to obtain \(7x = 14\). Then, divide both sides by 7, resulting in \(x = 2\).

Key Concepts

Powers of 2

Working with powers of 2 can greatly simplify solving exponential equations. In this problem, we deal with numbers like 8 and 64, which can be rewritten as powers of 2. This step is crucial because it transforms seemingly complex calculations into simpler operations that follow the rules of exponents.

• The number 8 can be expressed as \(8 = 2^3\).
• Similarly, 64 can be expressed as \(64 = 2^6\).

Rewriting these numbers in terms of powers of 2 ensures that both sides of our equation are consistent, making it easier to apply the properties of exponents. This method is especially useful in equations where exponents have variables, allowing you to solve them uniformly.

Properties of Exponents

Understanding the properties of exponents is essential for simplifying exponential expressions. These rules allow us to operate on expressions with the same base effectively. Two critical properties used in this exercise include:

• Product of Powers Property: When multiplying two exponents with the same base, like \(a^m\) and \(a^n\), you can simplify them by adding the exponents. The expression becomes \(a^{m+n}\).
• Power of a Power Property: When an exponent is raised to another exponent, such as \((a^m)^n\), you multiply the exponents: \(a^{m \cdot n}\).

By applying these properties, you can transform complex exponential terms into simpler ones, paving the way for easier equation solving. Use these rules to combine and simplify exponents efficiently.

Solving Equations

The process of solving exponential equations often involves making the equation look simpler by using the same base for all terms whenever possible. Here's a short guide on how to approach such problems:

• First, express all numbers as powers of a common base, such as 2 in our exercise.
• Then, simplify the left side using the properties of exponents by combining the exponents.
• With the equation simplified to a form \(2^a = 2^b\), set the exponents equal since they share the same base.
• Finally, solve the resulting linear equation for the variable.

This method standardizes the problem, allowing for a clear pathway to find the solution, by capitalizing on the consistent nature of exponent rules.

Step-by-Step Solution

Providing a detailed step-by-step breakdown assists in breaking down the process into digestible parts:

• Step 1: Convert all terms to the same base, here powers of 2, transforming the given equation \((2^{x+1})(8^{2x-3}) = 64\) into \((2^{x+1})(2^{3(2x-3)}) = 2^6\).
• Step 2: Use the product of powers property to simplify the left side, resulting in \(2^{x+1+6x-9}\).
• Step 3: Combine and simplify the exponents to get \((x+1) + (6x-9) = 7x - 8\). The equation becomes \(2^{7x-8} = 2^6\).
• Step 4: Since the bases are the same on both sides, equate the exponents: \(7x - 8 = 6\).
• Step 5: Solve for \(x\) by isolating the variable. Add 8 to both sides to obtain \(7x = 14\), then divide by 7, resulting in \(x = 2\).

Following each step carefully ensures clarity and understanding of how exponential equations can be methodically solved.