Welcome to Westonci.ca, the Q&A platform where your questions are met with detailed answers from experienced experts. Get immediate and reliable answers to your questions from a community of experienced professionals on our platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
To solve the equation [tex]\(9^{x-3} = 729\)[/tex], let's follow these steps:
1. Recognize the base relationships:
- Notice that 9 can be written as [tex]\(3^2\)[/tex].
- Notice that 729 can be written as [tex]\(3^6\)[/tex].
2. Rewrite the equation using powers of 3:
[tex]\[ 9^{x-3} = 729 \][/tex]
[tex]\[ (3^2)^{x-3} = 3^6 \][/tex]
3. Simplify the left-hand side:
When we raise a power to another power in exponents, we multiply the exponents.
[tex]\[ 3^{2(x-3)} = 3^6 \][/tex]
4. Set the exponents equal to each other:
Since the bases are the same, we can set the exponents equal to each other.
[tex]\[ 2(x-3) = 6 \][/tex]
5. Solve for [tex]\(x\)[/tex]:
- Distribute the 2 on the left-hand side:
[tex]\[ 2x - 6 = 6 \][/tex]
- Add 6 to both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[ 2x = 6 + 6 \][/tex]
[tex]\[ 2x = 12 \][/tex]
- Divide by 2 to solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{12}{2} \][/tex]
[tex]\[ x = 6 \][/tex]
So, the solution to the equation [tex]\(9^{x-3} = 729\)[/tex] is:
[tex]\[ x = 6 \][/tex]
1. Recognize the base relationships:
- Notice that 9 can be written as [tex]\(3^2\)[/tex].
- Notice that 729 can be written as [tex]\(3^6\)[/tex].
2. Rewrite the equation using powers of 3:
[tex]\[ 9^{x-3} = 729 \][/tex]
[tex]\[ (3^2)^{x-3} = 3^6 \][/tex]
3. Simplify the left-hand side:
When we raise a power to another power in exponents, we multiply the exponents.
[tex]\[ 3^{2(x-3)} = 3^6 \][/tex]
4. Set the exponents equal to each other:
Since the bases are the same, we can set the exponents equal to each other.
[tex]\[ 2(x-3) = 6 \][/tex]
5. Solve for [tex]\(x\)[/tex]:
- Distribute the 2 on the left-hand side:
[tex]\[ 2x - 6 = 6 \][/tex]
- Add 6 to both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[ 2x = 6 + 6 \][/tex]
[tex]\[ 2x = 12 \][/tex]
- Divide by 2 to solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{12}{2} \][/tex]
[tex]\[ x = 6 \][/tex]
So, the solution to the equation [tex]\(9^{x-3} = 729\)[/tex] is:
[tex]\[ x = 6 \][/tex]
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.