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Chapter 4: Problem 26

Solve the following equations for \(x\) \(\left(3^{2 x} \cdot 3^{2}\right)^{4}=3\)

Short Answer

Expert verified
The solution for \(x\) is \ x = -\frac{7}{8} \.

Step by step solution

01

Simplify the left-hand side using exponent rules

The left-hand side of the equation is \(\big(3^{2x} \times 3^2\big)^4\). Use the rule \(a^m \times a^n = a^{m+n}\) to combine the exponents of 3 inside the parentheses to get \(3^{2x + 2}\).
02

Apply the power rule to simplify further

Now, apply the power rule \( (a^m)^n = a^{m \times n} \) to the expression \(3^{2x+2}\), raised to the power of 4. This gives \(\big(3^{2x+2}\big)^4 = 3^{(2x+2) \times 4} = 3^{8x+8}\).
03

Set the simplified expression equal to the right-hand side

We now have the equation \(3^{8x + 8} = 3^1\). Since the bases are the same, we can set the exponents equal to each other: \(8x + 8 = 1\).
04

Solve for \(x\)

Solve the equation \(8x + 8 = 1\) by isolating \(x\). Subtract 8 from both sides to get \(8x = -7\). Then divide by 8: \ x = -\frac{7}{8} \.

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Key Concepts

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

Exponent Rules
Exponent rules are essential when dealing with exponential equations. They help simplify and solve expressions with the same base. The two key rules used in this exercise are:

  • The Product Rule: When you multiply two exponents with the same base, you add their powers: \(a^m \times a^n = a^{m+n}\). In this exercise, we combined \(3^{2x}\) and \(3^2\) inside the parentheses by adding their exponents to get \(3^{2x + 2}\).
  • The Power Rule: When raising an exponent to another power, you multiply the exponents: \((a^m)^n = a^{m \times n}\). Here, we applied this rule to \(3^{2x+2}\), which was raised to the power of 4, resulting in \(3^{(2x+2) \times 4} = 3^{8x + 8}\).
Understanding these rules will help you manage and simplify exponential expressions effectively.
Simplifying Expressions
Simplifying expressions step-by-step can make solving complex equations easier. In our problem, we started with \(\big(3^{2x} \times 3^2\big)^4\). First, we used the product rule to combine the exponents inside the parentheses to get \(3^{2x + 2}\).

Next, we applied the power rule to this new expression, raising \(3^{2x + 2}\) to the power of 4, resulting in \(3^{8x + 8}\).

By systematically applying these rules, we reduced the complex expression into a simpler form, making the equation easier to handle in the subsequent steps. This approach can be applied to other exponential equations for more efficient problem-solving.
Solving Equations
Once we've simplified the expression, the next step is solving the equation. We set our simplified left-hand side equal to the right-hand side: \(3^{8x + 8} = 3^1\). Since both sides have the same base (3), we can set the exponents equal to each other: \(8x + 8 = 1\)

To solve for \x\, we first isolate \x\ by subtracting 8 from both sides, yielding \(8x = -7\). Next, we divide by 8 to find \(x\): \x = -\frac{7}{8}\

This method is a straightforward way to solve exponential equations with the same base. Applying these steps ensures we can systematically find the solution to x.

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Most popular questions from this chapter

Solve the given equation for \(x .\) \(\ln x^{2}-\ln 2 x+1=0\)

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Solve the following equations for \(x .\) \(4-\ln x=0\)

(a) Find the point on the graph of \(y=e^{-x}\) where the tangent line has slope \(-2\). (b) Plot the graphs of \(y=e^{-x}\) and the tangent line in part (a).

Differentiate the following functions. \(y=\ln \left(\frac{x-1}{x+1}\right)\)
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