Find the best solutions to your questions at Westonci.ca, the premier Q&A platform with a community of knowledgeable experts. Discover in-depth solutions to your questions from a wide range of experts on our user-friendly Q&A platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
To solve the given equation [tex]\( 64 = 32^{x+1} \)[/tex], let's follow a step-by-step approach:
1. Rewrite the bases in terms of powers of 2:
- Notice that [tex]\( 32 \)[/tex] can be written as [tex]\( 2^5 \)[/tex].
- Also, [tex]\( 64 \)[/tex] can be written as [tex]\( 2^6 \)[/tex].
2. Express [tex]\( 32^{x+1} \)[/tex] in terms of base 2:
- Given [tex]\( 32 = 2^5 \)[/tex], it follows that [tex]\( 32^{x+1} = (2^5)^{x+1} \)[/tex].
- Using the property of exponents [tex]\((a^m)^n = a^{mn}\)[/tex], we get:
[tex]\[ (2^5)^{x+1} = 2^{5(x+1)}. \][/tex]
3. Rewrite the equation using these expressions:
- Now the equation [tex]\( 64 = 32^{x+1} \)[/tex] becomes:
[tex]\[ 2^6 = 2^{5(x+1)}. \][/tex]
4. Set the exponents equal:
- Since the bases (2) are the same, we can equate the exponents:
[tex]\[ 6 = 5(x+1). \][/tex]
5. Solve for [tex]\( x \)[/tex]:
- First, distribute the 5 on the right-hand side:
[tex]\[ 6 = 5x + 5. \][/tex]
- Next, subtract 5 from both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[ 6 - 5 = 5x. \][/tex]
[tex]\[ 1 = 5x. \][/tex]
- Finally, divide both sides by 5 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{1}{5}. \][/tex]
Thus, the solution is:
[tex]\[ x = 0.2. \][/tex]
1. Rewrite the bases in terms of powers of 2:
- Notice that [tex]\( 32 \)[/tex] can be written as [tex]\( 2^5 \)[/tex].
- Also, [tex]\( 64 \)[/tex] can be written as [tex]\( 2^6 \)[/tex].
2. Express [tex]\( 32^{x+1} \)[/tex] in terms of base 2:
- Given [tex]\( 32 = 2^5 \)[/tex], it follows that [tex]\( 32^{x+1} = (2^5)^{x+1} \)[/tex].
- Using the property of exponents [tex]\((a^m)^n = a^{mn}\)[/tex], we get:
[tex]\[ (2^5)^{x+1} = 2^{5(x+1)}. \][/tex]
3. Rewrite the equation using these expressions:
- Now the equation [tex]\( 64 = 32^{x+1} \)[/tex] becomes:
[tex]\[ 2^6 = 2^{5(x+1)}. \][/tex]
4. Set the exponents equal:
- Since the bases (2) are the same, we can equate the exponents:
[tex]\[ 6 = 5(x+1). \][/tex]
5. Solve for [tex]\( x \)[/tex]:
- First, distribute the 5 on the right-hand side:
[tex]\[ 6 = 5x + 5. \][/tex]
- Next, subtract 5 from both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[ 6 - 5 = 5x. \][/tex]
[tex]\[ 1 = 5x. \][/tex]
- Finally, divide both sides by 5 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{1}{5}. \][/tex]
Thus, the solution is:
[tex]\[ x = 0.2. \][/tex]
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.