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Chapter 4: Problem 25

Solve the following equations for \(x\). $$\left(2^{x+1} \cdot 2^{-3}\right)^{2}=2$$

Short Answer

Expert verified
x = \frac{5}{2}.

Step by step solution

01

- Simplify the expression inside the parentheses

Combine the terms inside the parentheses using the properties of exponents: \(2^{x+1} \times 2^{-3} = 2^{(x+1) + (-3)} = 2^{x-2}\)
02

- Apply the exponentiation

Raise the simplified expression to the power of 2: \[\big(2^{x-2}\big)^2 = 2^{2(x-2)} = 2^{2x-4}\]
03

- Set the exponent equal to the other side of the equation

Since the bases are the same, set the exponents equal to each other: \[2x - 4 = 1\]
04

- Solve for x

Isolate \(x\): \[2x - 4 = 1\]Add 4 to both sides: \[2x = 5\] Divide by 2: \[x = \frac{5}{2}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

properties of exponents
Understanding the properties of exponents is crucial for solving exponential equations. These properties help us simplify and manipulate expressions to find solutions.
For example, when you multiply two exponential terms with the same base, you add the exponents: \(a^m \times a^n = a^{m+n}\).
Similarly, dividing exponential terms means subtracting the exponents: \(a^m 梅 a^n = a^{m-n}\).
Another important property is the power of a power rule. When you raise an exponent to another exponent, you multiply them: \( (a^m)^n = a^{m \times n}\).
In the given problem, we used these properties to simplify the term inside the parentheses.
solving for variables
Solving for variables in exponential equations involves isolating the variable. By putting the equation in a simpler form, it becomes easier to solve.
After using the properties of exponents, the equation becomes straightforward, often resulting in simple linear equations.
For example, in this exercise, we simplified \(2^{x-2}\) to \(2^{2(x-2)}\.\)
Once simplified, it was clear that the exponents must equal each other, allowing us to set up an easy-to-solve equation: \(2x - 4 = 1\).
Always remember, manipulating terms to isolate your variable is essential for finding the answer.
algebraic manipulation
Algebraic manipulation refers to the series of steps taken to rearrange and simplify equations. It's like solving a puzzle, with each step bringing you closer to the solution.
In this problem, several key manipulations were performed:
  • Combining exponents inside the parentheses: \(2^{(x+1) + (-3)} = 2^{x-2}\)
  • Applying the power of a power rule: \( (2^{x-2})^2 = 2^{2(x-2)} = 2^{2x-4}\)
Finally, we set the exponent equal to the exponent on the other side of the equation and solved: \(2x - 4 = 1\). By adding 4 to both sides and dividing by 2, we found \(x = \frac{5}{2}\).
Perfecting these steps helps to systematically solve even more complex equations.

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