5^21 x 4^11 = 2 x 10^n

[smritihemdev]

5^21 x 4^11 = 2 x 10^n

What is n?

[venkat_yj]

5^21 x 4^11 = 2 x 10^n

LHS
5^21 x (2^2)^11
5^21 x 2^22

RHS
2 x 10^n
2 x (2x5)^n
2 x (2^n) x (5^n)
2^(n+1) x (5^n)

Now given LHS=RHS
n = 21

[Ben Ku]

When you encounter these problems, you want to use the rules of exponents to equate exponents with the same bases.

We can manipulate the left side:

5^21 * 4^11
= 5^21 * (2^2)^11
= 5^21 * 2^22 because (a^m)^n = a^m*n
= 5^21 * 2^21 * 2^1 because a^m * a^n = a^(m+n)
= (5*2)^21 * 2 because a^m * b^m = (ab)^m
= 2 * 10^21

It's clear that n = 21.