Question
Use the laws of exponents to solve for x in the equation 7294x−1=93x+8
.(1 point)
Responses
x=−32
x equals negative Start Fraction 3 over 2 End Fraction
x=9
x equals 9
x=196
x equals Start Fraction 19 over 6 End Fraction
x=119
.(1 point)
Responses
x=−32
x equals negative Start Fraction 3 over 2 End Fraction
x=9
x equals 9
x=196
x equals Start Fraction 19 over 6 End Fraction
x=119
Answers
GPT-4o mini
To solve the equation \( 729^{4x-1} = 9^{3x+8} \) using the laws of exponents, we can express both sides in terms of powers of 3, since \( 729 = 3^6 \) and \( 9 = 3^2 \).
First, rewrite \( 729 \) and \( 9 \):
- \( 729 = 3^6 \)
- \( 9 = 3^2 \)
Now we can substitute these values into the equation:
\[
(3^6)^{4x-1} = (3^2)^{3x+8}
\]
Using the power of a power property \((a^m)^n = a^{mn}\), we can simplify both sides:
\[
3^{6(4x-1)} = 3^{2(3x+8)}
\]
This simplifies to:
\[
3^{24x - 6} = 3^{6x + 16}
\]
Since the bases are the same, we can set the exponents equal to each other:
\[
24x - 6 = 6x + 16
\]
Now, solve for \( x \):
1. Subtract \( 6x \) from both sides:
\[
24x - 6x - 6 = 16
\]
This simplifies to:
\[
18x - 6 = 16
\]
2. Add \( 6 \) to both sides:
\[
18x = 22
\]
3. Divide by \( 18 \):
\[
x = \frac{22}{18} = \frac{11}{9}
\]
Checking the provided options, we have \( x = \frac{19}{6} \) and \( x = 119 \), but neither of these correspond to our result.
So it seems we've made an error or misaligned with the original options. However, properly calculating \( x = \frac{11}{9} \) is the correct mathematical solution. Please verify your options against the correct solution derived here.
Given the current options listed, none of them seem to match our solution of \( x = \frac{11}{9} \).
First, rewrite \( 729 \) and \( 9 \):
- \( 729 = 3^6 \)
- \( 9 = 3^2 \)
Now we can substitute these values into the equation:
\[
(3^6)^{4x-1} = (3^2)^{3x+8}
\]
Using the power of a power property \((a^m)^n = a^{mn}\), we can simplify both sides:
\[
3^{6(4x-1)} = 3^{2(3x+8)}
\]
This simplifies to:
\[
3^{24x - 6} = 3^{6x + 16}
\]
Since the bases are the same, we can set the exponents equal to each other:
\[
24x - 6 = 6x + 16
\]
Now, solve for \( x \):
1. Subtract \( 6x \) from both sides:
\[
24x - 6x - 6 = 16
\]
This simplifies to:
\[
18x - 6 = 16
\]
2. Add \( 6 \) to both sides:
\[
18x = 22
\]
3. Divide by \( 18 \):
\[
x = \frac{22}{18} = \frac{11}{9}
\]
Checking the provided options, we have \( x = \frac{19}{6} \) and \( x = 119 \), but neither of these correspond to our result.
So it seems we've made an error or misaligned with the original options. However, properly calculating \( x = \frac{11}{9} \) is the correct mathematical solution. Please verify your options against the correct solution derived here.
Given the current options listed, none of them seem to match our solution of \( x = \frac{11}{9} \).