### Solving Exponential Equations Using a Common Base #### Problem Statement: Solve the equation \(4^{x+3} = 2^{3x + 1}\) by using a common base. Enter your answer in the box provided. \[ \text{x = } \_\_\_\_\_ \] --- #### Solution Approach: 1. **Rewrite in Terms of a Common Base:** - Notice that 4 is \(2^2\). Thus, the equation \(4^{x+3}\) can be rewritten as \((2^2)^{x+3}\). - Rewrite the equation using the common base: \( (2^2)^{x+3} = 2^{3x + 1} \). 2. **Simplify the Exponents:** - Apply the power of a power property \((a^m)^n = a^{mn}\): \[ 2^{2(x+3)} = 2^{3x + 1} \] - Simplify the left side: \[ 2^{2x + 6} = 2^{3x + 1} \] 3. **Set the Exponents Equal:** - Since the bases are the same (base 2), you can set the exponents equal to each other: \[ 2x + 6 = 3x + 1 \] 4. **Solve for x:** - Isolate \(x\) by moving all \(x\)-terms to one side: \[ 2x + 6 - 3x = 1 \] \[ -x + 6 = 1 \] \[ -x = 1 - 6 \] \[ -x = -5 \] \[ x = 5 \] #### Answer: \[ x = 5 \] --- Enter your answer in the box: \[ x = 5 \] This equation solving method demonstrates how to manipulate and solve exponential equations by expressing them with a common base. Use these steps to simplify and find the solution efficiently.
### Solving Exponential Equations Using a Common Base #### Problem Statement: Solve the equation \(4^{x+3} = 2^{3x + 1}\) by using a common base. Enter your answer in the box provided. \[ \text{x = } \_\_\_\_\_ \] --- #### Solution Approach: 1. **Rewrite in Terms of a Common Base:** - Notice that 4 is \(2^2\). Thus, the equation \(4^{x+3}\) can be rewritten as \((2^2)^{x+3}\). - Rewrite the equation using the common base: \( (2^2)^{x+3} = 2^{3x + 1} \). 2. **Simplify the Exponents:** - Apply the power of a power property \((a^m)^n = a^{mn}\): \[ 2^{2(x+3)} = 2^{3x + 1} \] - Simplify the left side: \[ 2^{2x + 6} = 2^{3x + 1} \] 3. **Set the Exponents Equal:** - Since the bases are the same (base 2), you can set the exponents equal to each other: \[ 2x + 6 = 3x + 1 \] 4. **Solve for x:** - Isolate \(x\) by moving all \(x\)-terms to one side: \[ 2x + 6 - 3x = 1 \] \[ -x + 6 = 1 \] \[ -x = 1 - 6 \] \[ -x = -5 \] \[ x = 5 \] #### Answer: \[ x = 5 \] --- Enter your answer in the box: \[ x = 5 \] This equation solving method demonstrates how to manipulate and solve exponential equations by expressing them with a common base. Use these steps to simplify and find the solution efficiently.
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
Related questions
Question
![### Solving Exponential Equations Using a Common Base
#### Problem Statement:
Solve the equation \(4^{x+3} = 2^{3x + 1}\) by using a common base. Enter your answer in the box provided.
\[ \text{x = } \_\_\_\_\_ \]
---
#### Solution Approach:
1. **Rewrite in Terms of a Common Base:**
- Notice that 4 is \(2^2\). Thus, the equation \(4^{x+3}\) can be rewritten as \((2^2)^{x+3}\).
- Rewrite the equation using the common base: \( (2^2)^{x+3} = 2^{3x + 1} \).
2. **Simplify the Exponents:**
- Apply the power of a power property \((a^m)^n = a^{mn}\):
\[ 2^{2(x+3)} = 2^{3x + 1} \]
- Simplify the left side:
\[ 2^{2x + 6} = 2^{3x + 1} \]
3. **Set the Exponents Equal:**
- Since the bases are the same (base 2), you can set the exponents equal to each other:
\[ 2x + 6 = 3x + 1 \]
4. **Solve for x:**
- Isolate \(x\) by moving all \(x\)-terms to one side:
\[ 2x + 6 - 3x = 1 \]
\[ -x + 6 = 1 \]
\[ -x = 1 - 6 \]
\[ -x = -5 \]
\[ x = 5 \]
#### Answer:
\[ x = 5 \]
---
Enter your answer in the box:
\[ x = 5 \]
This equation solving method demonstrates how to manipulate and solve exponential equations by expressing them with a common base. Use these steps to simplify and find the solution efficiently.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbc2f1e2b-bc72-4ede-9c19-0c442dcb26d6%2F8c4e5614-4edb-4466-8360-2d109e361110%2Fo9ul00e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving Exponential Equations Using a Common Base
#### Problem Statement:
Solve the equation \(4^{x+3} = 2^{3x + 1}\) by using a common base. Enter your answer in the box provided.
\[ \text{x = } \_\_\_\_\_ \]
---
#### Solution Approach:
1. **Rewrite in Terms of a Common Base:**
- Notice that 4 is \(2^2\). Thus, the equation \(4^{x+3}\) can be rewritten as \((2^2)^{x+3}\).
- Rewrite the equation using the common base: \( (2^2)^{x+3} = 2^{3x + 1} \).
2. **Simplify the Exponents:**
- Apply the power of a power property \((a^m)^n = a^{mn}\):
\[ 2^{2(x+3)} = 2^{3x + 1} \]
- Simplify the left side:
\[ 2^{2x + 6} = 2^{3x + 1} \]
3. **Set the Exponents Equal:**
- Since the bases are the same (base 2), you can set the exponents equal to each other:
\[ 2x + 6 = 3x + 1 \]
4. **Solve for x:**
- Isolate \(x\) by moving all \(x\)-terms to one side:
\[ 2x + 6 - 3x = 1 \]
\[ -x + 6 = 1 \]
\[ -x = 1 - 6 \]
\[ -x = -5 \]
\[ x = 5 \]
#### Answer:
\[ x = 5 \]
---
Enter your answer in the box:
\[ x = 5 \]
This equation solving method demonstrates how to manipulate and solve exponential equations by expressing them with a common base. Use these steps to simplify and find the solution efficiently.
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