How would you solve this?
if 5^11 x 4^5 = 5 x 10^k, what is the value of k?
integers and number properties
This topic has expert replies
-
- Senior | Next Rank: 100 Posts
- Posts: 92
- Joined: Thu Oct 06, 2011 8:06 am
- Thanked: 18 times
this can also be written as 5 x 5^10 x 4^55^11 x 4^5 = 5 x 10^k
or 5 x 5^10 x (2^2)^5
or 5 x 5^10 x 2^10
or 5 x (5 x 2)^10
or 5 x 10^10
thus k=10
- tpr-becky
- GMAT Instructor
- Posts: 509
- Joined: Wed Apr 21, 2010 1:08 pm
- Location: Irvine, CA
- Thanked: 199 times
- Followed by:85 members
- GMAT Score:750
realize that you can multiply distinct bases that have the same exponent together and get a new number that has the same exponent - for example - 2^2(3^2) = 6^2
Now to get to a 10 as a base you need a 5 and a 2 - you can work with the first part of the equation or the second - since Neo worked with the first, I'll show you how to work with the second.
5 x 10^k = 5 x 5^k x 2^k = 5^(k+1) x 4^k = 5^(k+1) x 4^(k/2)
On the other side we have 5^11 x 4^5 so we know k+1= 11 and k/2=5 thus for each k = 10.
The skill involved here is factoring and multiplying bases
Now to get to a 10 as a base you need a 5 and a 2 - you can work with the first part of the equation or the second - since Neo worked with the first, I'll show you how to work with the second.
5 x 10^k = 5 x 5^k x 2^k = 5^(k+1) x 4^k = 5^(k+1) x 4^(k/2)
On the other side we have 5^11 x 4^5 so we know k+1= 11 and k/2=5 thus for each k = 10.
The skill involved here is factoring and multiplying bases
Becky
Master GMAT Instructor
The Princeton Review
Irvine, CA
Master GMAT Instructor
The Princeton Review
Irvine, CA
GMAT/MBA Expert
- Anurag@Gurome
- GMAT Instructor
- Posts: 3835
- Joined: Fri Apr 02, 2010 10:00 pm
- Location: Milpitas, CA
- Thanked: 1854 times
- Followed by:523 members
- GMAT Score:770
Given: 5^11 x 4^5 = 5 x 10^kjzebra10 wrote:How would you solve this?
if 5^11 x 4^5 = 5 x 10^k, what is the value of k?
Now 5 x 10^k = 5 x (2 * 5)^k = 2^k * 5^(k + 1)
Also, 4^5 = (2²)^5 = 2^10
So, 2^10 * 5^11 = 2^k * 5^(k + 1)
Now it can be seen that bases are the same on both sides, so equating the exponents, we get k = 10
Anurag Mairal, Ph.D., MBA
GMAT Expert, Admissions and Career Guidance
Gurome, Inc.
1-800-566-4043 (USA)
Join Our Facebook Groups
GMAT with Gurome
https://www.facebook.com/groups/272466352793633/
Admissions with Gurome
https://www.facebook.com/groups/461459690536574/
Career Advising with Gurome
https://www.facebook.com/groups/360435787349781/
GMAT Expert, Admissions and Career Guidance
Gurome, Inc.
1-800-566-4043 (USA)
Join Our Facebook Groups
GMAT with Gurome
https://www.facebook.com/groups/272466352793633/
Admissions with Gurome
https://www.facebook.com/groups/461459690536574/
Career Advising with Gurome
https://www.facebook.com/groups/360435787349781/