Question

which equation is equivalent to (16^{2n} = 32^{n+3})?

• (8^{4n} = 8^{4n+3})
• (8^{4n} = 8^{4n+12})
• (2^{8n} = 2^{5n+15})
• (2^{8n} = 2^{5n+3})

Explanation:

⚡ Using what you learned: Solving Exponential Equations

Step 1: Identify a common base

Both bases, \(16\) and \(32\), can be written as powers of \(2\):

$$ 16 = 2^4 $$

$$ 32 = 2^5 $$

Step 2: Substitute and apply exponent rules

Substitute these bases back into the original equation \(16^{2n} = 32^{n+3}\):

$$ (2^4)^{2n} = (2^5)^{n+3} $$

Apply the power of a power rule, \((a^m)^n = a^{m \cdot n}\):

$$ 2^{4 \cdot 2n} = 2^{5(n+3)} $$

$$ 2^{8n} = 2^{5n+15} $$

Answer:

\(2^{8n} = 2^{5n+15}\)