### Solving Exponential Equations **Problem Statement:** Solve the following exponential equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places. \[ 9^{x - 6} = 81 \] **Steps to Solve:** 1. Recognize that both sides of the equation can be expressed as powers of 3. 2. Rewrite \( 9 \) as \( 3^2 \) and \( 81 \) as \( 3^4 \). Thus, the equation becomes: \[ (3^2)^{x - 6} = 3^4 \] 3. Apply the power rule \((a^m)^n = a^{mn}\). This simplifies to: \[ 3^{2(x - 6)} = 3^4 \] 4. Since the bases are the same, set the exponents equal to each other. \[2(x - 6) = 4\] 5. Solve the equation for \(x\). \[2x - 12 = 4\] \[2x = 16\] \[x = 8\] **Exact Answer:** - Select the correct choice below and, if necessary, fill in the answer box to complete your choice. - \[ \text{A.} \] The solution set is \( \boxed{8} \). (Simplify your answer. Type an exact answer.) - \[ \text{B.} \] There is no solution. **Answer Rounded to Three Decimal Places:** - Select the correct choice below and, if necessary, fill in the answer box to complete your choice. - \[ \text{A.} \] The solution set is \( \boxed{8.000} \). (Simplify your answer. Type an integer or decimal rounded to three decimal places as needed.) - \[ \text{B.} \] There is no solution.
### Solving Exponential Equations **Problem Statement:** Solve the following exponential equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places. \[ 9^{x - 6} = 81 \] **Steps to Solve:** 1. Recognize that both sides of the equation can be expressed as powers of 3. 2. Rewrite \( 9 \) as \( 3^2 \) and \( 81 \) as \( 3^4 \). Thus, the equation becomes: \[ (3^2)^{x - 6} = 3^4 \] 3. Apply the power rule \((a^m)^n = a^{mn}\). This simplifies to: \[ 3^{2(x - 6)} = 3^4 \] 4. Since the bases are the same, set the exponents equal to each other. \[2(x - 6) = 4\] 5. Solve the equation for \(x\). \[2x - 12 = 4\] \[2x = 16\] \[x = 8\] **Exact Answer:** - Select the correct choice below and, if necessary, fill in the answer box to complete your choice. - \[ \text{A.} \] The solution set is \( \boxed{8} \). (Simplify your answer. Type an exact answer.) - \[ \text{B.} \] There is no solution. **Answer Rounded to Three Decimal Places:** - Select the correct choice below and, if necessary, fill in the answer box to complete your choice. - \[ \text{A.} \] The solution set is \( \boxed{8.000} \). (Simplify your answer. Type an integer or decimal rounded to three decimal places as needed.) - \[ \text{B.} \] There is no solution.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![### Solving Exponential Equations
**Problem Statement:**
Solve the following exponential equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places.
\[ 9^{x - 6} = 81 \]
**Steps to Solve:**
1. Recognize that both sides of the equation can be expressed as powers of 3.
2. Rewrite \( 9 \) as \( 3^2 \) and \( 81 \) as \( 3^4 \).
Thus, the equation becomes:
\[ (3^2)^{x - 6} = 3^4 \]
3. Apply the power rule \((a^m)^n = a^{mn}\).
This simplifies to:
\[ 3^{2(x - 6)} = 3^4 \]
4. Since the bases are the same, set the exponents equal to each other.
\[2(x - 6) = 4\]
5. Solve the equation for \(x\).
\[2x - 12 = 4\]
\[2x = 16\]
\[x = 8\]
**Exact Answer:**
- Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
- \[ \text{A.} \] The solution set is \( \boxed{8} \).
(Simplify your answer. Type an exact answer.)
- \[ \text{B.} \] There is no solution.
**Answer Rounded to Three Decimal Places:**
- Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
- \[ \text{A.} \] The solution set is \( \boxed{8.000} \).
(Simplify your answer. Type an integer or decimal rounded to three decimal places as needed.)
- \[ \text{B.} \] There is no solution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc6811bcd-02f8-46fa-b044-8f44b9c4930d%2Fa898b658-4388-4d12-a58d-0c689190c481%2Fdrb3ik.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving Exponential Equations
**Problem Statement:**
Solve the following exponential equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places.
\[ 9^{x - 6} = 81 \]
**Steps to Solve:**
1. Recognize that both sides of the equation can be expressed as powers of 3.
2. Rewrite \( 9 \) as \( 3^2 \) and \( 81 \) as \( 3^4 \).
Thus, the equation becomes:
\[ (3^2)^{x - 6} = 3^4 \]
3. Apply the power rule \((a^m)^n = a^{mn}\).
This simplifies to:
\[ 3^{2(x - 6)} = 3^4 \]
4. Since the bases are the same, set the exponents equal to each other.
\[2(x - 6) = 4\]
5. Solve the equation for \(x\).
\[2x - 12 = 4\]
\[2x = 16\]
\[x = 8\]
**Exact Answer:**
- Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
- \[ \text{A.} \] The solution set is \( \boxed{8} \).
(Simplify your answer. Type an exact answer.)
- \[ \text{B.} \] There is no solution.
**Answer Rounded to Three Decimal Places:**
- Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
- \[ \text{A.} \] The solution set is \( \boxed{8.000} \).
(Simplify your answer. Type an integer or decimal rounded to three decimal places as needed.)
- \[ \text{B.} \] There is no solution.
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