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Chapter 11: Problem 69

Solve the given problems. Solve for \(x: 2^{5 x}=2^{7}\left(2^{2 x}\right)^{2}\).

Short Answer

Expert verified
The solution is \(x = 7\).

Step by step solution

01

Simplify the Right Side of the Equation

Start by simplifying the expression on the right side of the equation: \[2^{7}(2^{2x})^2 = 2^{7} \times 2^{4x}.\]This is because \[(2^{2x})^2 = 2^{2x \times 2} = 2^{4x}.\]
02

Combine and Equate the Exponents

Now the equation is \[2^{5x} = 2^{7 + 4x}.\]Since the bases are the same (base 2), we equate the exponents:\[5x = 7 + 4x.\]
03

Solve the Equation for x

Subtract \(4x\) from both sides to isolate \(x\): \[5x - 4x = 7.\]This simplifies to \[x = 7.\]

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Key Concepts

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

Simplifying Exponents
Simplifying exponents is a crucial part of understanding and solving exponential equations. When you see an expression with exponents, like \((2^{2x})^2\), you need to simplify it before proceeding further. This process involves using the power of a power rule, which states that \((a^m)^n = a^{m \times n}\).
In our example, \((2^{2x})^2\) simplifies to \(2^{2x \times 2}\), which equals \(2^{4x}\). Breaking this down:
  • Find the inner exponent: Here it's \(2x\).
  • Multiply it by the outer exponent: \(2x \times 2 = 4x\).
This makes handling exponential terms much easier, setting the stage for solving the equation by ensuring its simplicity.
Equation Solving
Equation solving, especially with exponential equations, involves several key steps aimed at isolating the variable you're solving for. In our equation \(2^{5x} = 2^{7 + 4x}\), the bases are the same, which simplifies the process.

Here's how you approach it:
  • The primary goal is to have one power of the base on each side of the equation.
  • Once both sides have the same base, you can directly equate the exponents, leading from \(2^{5x} = 2^{7 + 4x}\) to \(5x = 7 + 4x\).
  • This simplification works due to the fundamental property that if \(a^m = a^n\), then \(m = n\).
  • Remember, consistency in handling bases efficiently simplifies the equation to standard algebraic terms.
The emphasis is on reducing the complexity by zeroing in on the exponents.
Algebraic Manipulation
Algebraic manipulation is the bridge that takes you from a complex equation to finding the solution. After setting the exponents equal, you have a simple linear equation \(5x = 7 + 4x\). Solving this requires straightforward algebraic movements:
  • Step 1: Subtract \(4x\) from both sides, yielding \(5x - 4x = 7\).
  • Step 2: Simplify to \(x = 7\).
The critical thought here is isolating the variable \(x\) on one side of the equation, converting the expression into a solvable form.
Utilizing operations like addition, subtraction, or factorization allows for breaking down hurdles in complex expressions, bringing clarity and facilitating precise solutions to algebraic problems.

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Most popular questions from this chapter

Combine the terms into a single fraction, but do not rationalize the denominators. $$\frac{x^{2}}{\sqrt{2 x+1}}+2 x \sqrt{2 x+1}$$

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