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

1 answer

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:

\[ 9x - 3 = 8 \]

  1. Add \( 3 \) to both sides:

\[ 9x = 11 \]

  1. 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.