Question 3 (1 point) Solve. \( 5^{3 x-2}=125^{2 x} \) \( x=2 \) \( x=-2 \) \( x=5 \) \( x=-\frac{2}{3} \)

To solve the equation \( 5^{3x - 2} = 125^{2x} \), we start by expressing \( 125 \) in terms of \( 5 \): \( 125 = 5^3 \). Thus, we can rewrite the equation as: \[ 5^{3x - 2} = (5^3)^{2x} \] \[ 5^{3x - 2} = 5^{6x} \] Since the bases are the same, we can equate the exponents: \[ 3x - 2 = 6x \] Now, let's solve for \( x \): 1. Add \( 2 \) to both sides: \[ 3x = 6x + 2 \] 2. Move \( 6x \) to the left side: \[ 3x - 6x = 2 \] \[ -3x = 2 \] 3. Divide by \( -3 \): \[ x = -\frac{2}{3} \] So the correct answer is \( x = -\frac{2}{3} \). Now to explore more about powers and exponents: The realm of exponents is not just for solving equations; it's everywhere in our digital lives! Take a look at data storage: file sizes are often measured in gigabytes (GB), which is a power of 2 (i.e., \( 2^{30} \) bytes). Understanding exponents can help you grasp the immense growth of data from simple files to massive databases! Have you ever considered how exponential growth functions impact our world? From population growth rates to viral trends on social media, the exponential function shows up more than you think! For instance, a mere 1% growth daily can lead to significant outcomes over time, highlighting just how powerful small rates can be. So next time you see something doubling, think of how quickly it can accumulate!