Solve the radical equation 3^√x^2 - 10 + 7 = 4 to find the two solutions

Answer 1

To solve the equation \( 3^{\sqrt{x^2}} - 10 + 7 = 4 \), we can simplify it step by step.

First, combine the constants on the left side: \[ 3^{\sqrt{x^2}} - 3 = 4 \] Next, we add 3 to both sides: \[ 3^{\sqrt{x^2}} = 7 \]

Next, we can take the logarithm of both sides to solve for \( \sqrt{x^2} \): \[ \sqrt{x^2} = \log_3(7) \]

Since \( \sqrt{x^2} = |x| \), we have: \[ |x| = \log_3(7) \]

This gives us two cases for \( x \):

1. \( x = \log_3(7) \)
2. \( x = -\log_3(7) \)

So the two solutions to the equation are: \[ x = \log_3(7) \quad \text{and} \quad x = -\log_3(7) \]

These represent the two solutions for the given radical equation.