Question
Use the laws of exponents to solve for x in the equation 7294x−1=93x+8 .(1 point)
Responses
x=9
x equals 9
x=196
x equals Start Fraction 19 over 6 End Fraction
x=119
x equals Start Fraction 11 over 6 End Fraction
x=−32
Responses
x=9
x equals 9
x=196
x equals Start Fraction 19 over 6 End Fraction
x=119
x equals Start Fraction 11 over 6 End Fraction
x=−32
Answers
GPT-4o mini
To solve the equation \( 729^{4x - 1} = 9^{3x + 8} \) using the laws of exponents, we need to express both sides of the equation with the same base.
First, we can rewrite the bases:
1. \( 729 \) can be written as \( 9^3 \) because \( 9^3 = 729 \).
2. Therefore, \( 729^{4x - 1} = (9^3)^{4x - 1} = 9^{3(4x - 1)} = 9^{12x - 3} \).
Now, our equation becomes:
\[
9^{12x - 3} = 9^{3x + 8}
\]
Since the bases on both sides are the same, we can set the exponents equal to each other:
\[
12x - 3 = 3x + 8
\]
Next, we'll solve for \( x \):
1. Subtract \( 3x \) from both sides:
\[
12x - 3x - 3 = 8
\]
This simplifies to:
\[
9x - 3 = 8
\]
2. Add \( 3 \) to both sides:
\[
9x = 11
\]
3. Finally, divide by \( 9 \):
\[
x = \frac{11}{9}
\]
The options given in the responses seem to be a little different from what we derived. However, \( x = \frac{11}{9} \) can be represented differently. Let's confirm the calculation:
The solution is indeed \( x = \frac{11}{9} \).
Checking the options:
- \( x = \frac{11}{6} \) is incorrect.
- \( x = 196 \) is incorrect.
- \( x = 9 \) is incorrect.
- \( x = -32 \) is also incorrect.
Thus, based on the laws applied, the correct solution is \( x = \frac{11}{9} \).
If \( x = \frac{11}{9} \) is not listed, it may require verification of the problem formulation or redrafting the options.
First, we can rewrite the bases:
1. \( 729 \) can be written as \( 9^3 \) because \( 9^3 = 729 \).
2. Therefore, \( 729^{4x - 1} = (9^3)^{4x - 1} = 9^{3(4x - 1)} = 9^{12x - 3} \).
Now, our equation becomes:
\[
9^{12x - 3} = 9^{3x + 8}
\]
Since the bases on both sides are the same, we can set the exponents equal to each other:
\[
12x - 3 = 3x + 8
\]
Next, we'll solve for \( x \):
1. Subtract \( 3x \) from both sides:
\[
12x - 3x - 3 = 8
\]
This simplifies to:
\[
9x - 3 = 8
\]
2. Add \( 3 \) to both sides:
\[
9x = 11
\]
3. Finally, divide by \( 9 \):
\[
x = \frac{11}{9}
\]
The options given in the responses seem to be a little different from what we derived. However, \( x = \frac{11}{9} \) can be represented differently. Let's confirm the calculation:
The solution is indeed \( x = \frac{11}{9} \).
Checking the options:
- \( x = \frac{11}{6} \) is incorrect.
- \( x = 196 \) is incorrect.
- \( x = 9 \) is incorrect.
- \( x = -32 \) is also incorrect.
Thus, based on the laws applied, the correct solution is \( x = \frac{11}{9} \).
If \( x = \frac{11}{9} \) is not listed, it may require verification of the problem formulation or redrafting the options.