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

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VJesus12: Legendary Member; Posts: 2276; Joined: Sat Oct 14, 2017 6:10 amFollowed by: 3 members

by VJesus12 » Thu Mar 18, 2021 12:55 pm

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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

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Source: Beat The GMAT — Problem Solving | Thu Mar 18, 2021 12:55 pm

swerve: Legendary Member; Posts: 2545; Joined: Sun Oct 29, 2017 2:04 pmFollowed by: 6 members

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

by swerve » Thu Mar 18, 2021 4:40 pm

VJesus12 wrote: ↑
Thu Mar 18, 2021 12:55 pm
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

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