Welcome to Westonci.ca, your ultimate destination for finding answers to a wide range of questions from experts. Discover detailed solutions to your questions from a wide network of experts on our comprehensive Q&A 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]\(64^{3x} = 512^{2x + 12}\)[/tex], we begin by expressing the bases (64 and 512) as powers of 2.
1. [tex]\(64 = 2^6\)[/tex], so [tex]\(64^{3x} = (2^6)^{3x} = 2^{18x}\)[/tex].
2. [tex]\(512 = 2^9\)[/tex], so [tex]\(512^{2x + 12} = (2^9)^{2x + 12} = 2^{9(2x + 12)}\)[/tex].
Now, the original equation [tex]\(64^{3x} = 512^{2x + 12}\)[/tex] becomes:
[tex]\[2^{18x} = 2^{9(2x + 12)}\][/tex]
Since the bases are both 2, we can set the exponents equal to each other:
[tex]\[18x = 9(2x + 12)\][/tex]
Next, distribute the 9 on the right-hand side:
[tex]\[18x = 18x + 108\][/tex]
Subtract [tex]\(18x\)[/tex] from both sides:
[tex]\[18x - 18x = 108\][/tex]
[tex]\[0 = 108\][/tex]
This results in a contradiction, as [tex]\(0\)[/tex] cannot equal [tex]\(108\)[/tex]. Therefore, there is no value of [tex]\(x\)[/tex] that satisfies the equation [tex]\(64^{3x} = 512^{2x + 12}\)[/tex].
As a result, the correct answer is:
[tex]\[ \boxed{\text{no solution}} \][/tex]
1. [tex]\(64 = 2^6\)[/tex], so [tex]\(64^{3x} = (2^6)^{3x} = 2^{18x}\)[/tex].
2. [tex]\(512 = 2^9\)[/tex], so [tex]\(512^{2x + 12} = (2^9)^{2x + 12} = 2^{9(2x + 12)}\)[/tex].
Now, the original equation [tex]\(64^{3x} = 512^{2x + 12}\)[/tex] becomes:
[tex]\[2^{18x} = 2^{9(2x + 12)}\][/tex]
Since the bases are both 2, we can set the exponents equal to each other:
[tex]\[18x = 9(2x + 12)\][/tex]
Next, distribute the 9 on the right-hand side:
[tex]\[18x = 18x + 108\][/tex]
Subtract [tex]\(18x\)[/tex] from both sides:
[tex]\[18x - 18x = 108\][/tex]
[tex]\[0 = 108\][/tex]
This results in a contradiction, as [tex]\(0\)[/tex] cannot equal [tex]\(108\)[/tex]. Therefore, there is no value of [tex]\(x\)[/tex] that satisfies the equation [tex]\(64^{3x} = 512^{2x + 12}\)[/tex].
As a result, the correct answer is:
[tex]\[ \boxed{\text{no solution}} \][/tex]
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.