Question

choose the correct equation.

\\( (7)^{3x} = (7^3)^{2x+1} \\)

\\( (7^2)^{3x} = (7^3)^{2x+1} \\)

\\( (7^2)^{3x} = (7^4)^{2x+1} \\)

Explanation:

⚡ Using what you learned: Solving Exponential Equations

Step 1: Identify the original bases

To find the correct equation, we need to express both sides of an exponential equation using a common base. Let's look at the bases in the options: \( 7 \), \( 7^2 \) (which is \( 49 \)), \( 7^3 \) (which is \( 343 \)), and \( 7^4 \) (which is \( 2401 \)).

The options represent transformations of an equation of the form:

$$ (49)^{3x} = (343)^{2x+1} $$

Step 2: Rewrite with a common base of 7

Rewrite the numbers \( 49\) and \( 343 \) as powers of \( 7 \):

$$ 49 = 7^2 $$

$$ 343 = 7^3 $$

Substitute these values back into the equation:

$$ (7^2)^{3x} = (7^3)^{2x+1} $$

Answer:

$$ (7^2)^{3x} = (7^3)^{2x+1} $$

(the second option)