If \(5^{21}\cdot 4^{11}=2\cdot 10^n,\) what is the value of \(n?\)

[VJesus12]

If \(5^{21}\cdot 4^{11}=2\cdot 10^n,\) what is the value of \(n?\)

A. 11
B. 21
C. 22
D. 23
E. 32

Answer: B

Source: GMAT Prep

[swerve]

If \(5^{21}\cdot 4^{11}=2\cdot 10^n,\) what is the value of \(n?\)

A. 11
B. 21
C. 22
D. 23
E. 32

Answer: B

Source: GMAT Prep

\begin{align*}
5^{21} \cdot 4^{11} &= 2\cdot 10^p \\
5^{21} \cdot 2^{22} &= 2 \cdot (2\cdot 5)^p \\
5^{21} \cdot 2^{22} &= 2 \cdot 2^p \cdot 5^p \\
5^{21} \cdot 2^{22} &= 2^{p+1} \cdot 5^p
\end{align*}

Now, we equate the exponents an get
\begin{align*}
p+1 &= 22\\
p &= 21
\end{align*}

Therefore, B