Question

if \\(8^{3b-1} = 8^{b+3}\\), what is the value of \\(b\\)?

-2
-1
1
2

Explanation:

⚡ Using what you learned: Solving Exponential Equations

Step 1: Express bases with a common base

$$ 8 = 2^3 $$

$$ 16 = 2^4 $$

Substitute these into the equation:

$$ (2^3)^{3b - 1} = (2^4)^{b + 3} $$

Step 2: Simplify exponents

$$ 2^{3(3b - 1)} = 2^{4(b + 3)} $$

$$ 3(3b - 1) = 4(b + 3) $$

$$ 9b - 3 = 4b + 12 $$

Step 3: Solve for \(b\)

$$ 5b = 15 $$

$$ b = 3 $$

Let's double-check the original image to ensure the exponents are read correctly.
The image shows:

$$ 8^{3b-1} = 16^{b+3} $$

Wait, is the right base \(16\) or \(8\)? It looks like \(16^{b+3}\) or \(8^{b+3}\)?
If it is \(8^{3b-1} = 8^{b+3}\):

$$ 3b - 1 = b + 3 \implies 2b = 4 \implies b = 2 $$

Looking closely at the image, the base on the right is indeed \(8\). Let's re-examine: "If \(8^{3b-1} = 8^{b+3}\), what is the value of \(b\)?".
Yes, both bases are \(8\).

Let's solve with base \(8\):

$$ 3b - 1 = b + 3 $$

$$ 2b = 4 $$

$$ b = 2 $$

This matches the option \(2\) in the multiple-choice list.

Answer:

2