Solve for n: 2^n 4^n=64^n-36.

Since 4=2^2, 4^n=2^2n. Since 64=2^6, 64^n-36=2^6(n-36). Thus, 2^n+2n=2^6(n-36) 3n=6n-216 So 3n=216 n=72.

jmc

algebra intermediate

Problem

Solve for

n

2^{n} \cdot 4^{n} = 6 4^{n - 36}

Solution

Since

4 = 2^{2}

4^{n} = 2^{2 n}

. Since

64 = 2^{6}

6 4^{n - 36} = 2^{6 (n - 36)}

. Thus,

2^{n + 2 n} = 2^{6 (n - 36)} \Rightarrow 3 n = 6 n - 216

3 n = 216 \Rightarrow n = 72

Final answer