Question

Solve for x in the equation 103x−10=(1100)6x−1 .(1 point) Responses x=3 x equals 3 x=−89 x equals negative Start Fraction 8 over 9 End Fraction x=35 x equals Start Fraction 3 over 5 End Fraction x=45 x equals Start Fraction 4 over 5 End Fraction

Answer 1

To solve the equation \( 10^{3x - 10} = (1100)^{6x - 1} \), we can start by rewriting \( 1100 \) in terms of a base we can more easily manage.

First, we rewrite \( 1100 \):
\[
1100 = 11 \cdot 100 = 11 \cdot 10^2
\]

Now rewrite the equation:
\[
10^{3x - 10} = (11 \cdot 10^2)^{6x - 1}
\]

Using the properties of exponents, this expands to:
\[
10^{3x - 10} = 11^{6x - 1} \cdot 10^{2(6x - 1)}
\]
\[
10^{3x - 10} = 11^{6x - 1} \cdot 10^{12x - 2}
\]

Now, since we have base \( 10 \) on both sides, we can equate the exponents for the \( 10 \) terms:
\[
3x - 10 = 12x - 2
\]

Next, we’ll solve for \( x \):
1. Rearranging gives:
\[
3x - 12x = -2 + 10
\]
\[
-9x = 8
\]
\[
x = -\frac{8}{9}
\]

Now, let's check which option matches:
**Responses:**
- \( x = 3 \)
- \( x = -89 \)
- \( x = -\frac{8}{9} \)
- \( x = 35 \)
- \( x = \frac{3}{5} \)
- \( x = 45 \)
- \( x = \frac{4}{5} \)

The solution we found \( x = -\frac{8}{9} \) matches the option \( x = -\frac{8}{9} \), confirming our solution.

Thus, the solution is:
\[
\boxed{-\frac{8}{9}}
\]