value of n

If 5^21 * 4^11 = 2 * 10^n what is the value of n?

5^21 * 4^11 = 5^21 * 2^22 = 5^21 * 2^21 * 2 = 10^21 * 2 = 2 * 10^n => n = 21

5^21 * 4^11 = 2 * 10^n
5^21 * 2^22 = 2 * 2^n*5^n
so, n must be 21

just see the power of 5;
to make both sides equal, we need to have same powers of 5 on both sides. here 10 5*2; so power of 10 must be equal to power of 5

This question is based on the rule that if you have two different exponential numbers with different bases but the same exponent you can multiply the bases and keep the original exponent.

For instance 2^2 * 3^2 is the same as 6^2.

In this question you have 5 and 4 as bases but they do not mulitply to the 10 on the other side of the equation. Therefore you will have to change the 4 to a 2 and multiply the exponent by 2. 4^11 = 2^22

But you only have 21 5's so you have to switch so that you will get the same exponent on each number so the equation must be turned into 5^21 * 2 * 2^21. now you can combine the 5^21 and the 2^21 to get 10^21 with the extra two. Once you understand this concept you can do the problem by realizing that all of the 5's are used and you can just look at the exponent on the 5 to find the answer.

Just to make it even more simpler:

5^21*4^11=5^21*(2^2)^11=5^21*2^22=2*5^n*2^n - divide both sides by 2

5^21*2^21=5^n*2^n - here you can clearly see that the exponent n is equal to 21.