Question

solve: \\(7 + 42 \cdot 3^{2-3a} = 14 \cdot 3^{-2a} + 7\\)

\\(a = -3\\)
\\(a = 0\\)
\\(a = 3\\)
no solution

Explanation:

⚡ Using what you learned: Solving Exponential Equations

Step 1: Simplify the equation

Subtract \( 7 \) from both sides of the equation:

$$ 42 \cdot 3^{2-3a} = 14 \cdot 3^{-2a} $$

Step 2: Divide by constants

Divide both sides by \( 14 \):

$$ 3 \cdot 3^{2-3a} = 3^{-2a} $$

Step 3: Combine exponents on the left side

Using the exponent rule \( x^m \cdot x^n = x^{m+n} \):

$$ 3^{1 + (2-3a)} = 3^{-2a} $$

$$ 3^{3-3a} = 3^{-2a} $$

Step 4: Equate the exponents

Since the bases are the same, set the exponents equal to each other:

$$ 3 - 3a = -2a $$

Step 5: Solve for \( a \)

Add \( 3a \) to both sides:

$$ 3 = a $$

$$ a = 3 $$

Answer:

\( a = 3 \)