# Exponents

Follow the Knowsys Method and remember to read the problem, identify the bottom line, assess your options, and attack the problem. Then loop back to check that you answered the right question. For the vast majority of problems, you do not need to look at the answer choices before this point.

What is the largest possible integer value of n for which $5^{n}$ divides into $50^{7}$?

The bottom line is easy to find here: n=?

Now assess your options. You could look at the answer choices and plug them in, calculate each product, and see whether $50^{7}$ can divide by it evenly. But there must be a faster way! This is an exponent problem, so think about your exponent rules. If you can get the bases to match, finding the appropriate value of n will be easy.

Fortunately, 50 is a multiple of 5. It is also a multiple of 25.

$50=2(5^{2})$

Therefore,

$50^{7}=(2(5^{2}))^{7}$

Now you can apply the distributive property and the exponent rules.

$50^{7}=2^{7}(5^{2^{7}})$

$50^{7}=2^{7}(5^{14})$

Now you know that $(5^{14})$ is a product of  $50^{7}$. There's not much you can do from here, so look at the answer choices.

(A) 2
(B) 7
(C) 9
(D) 10
(E) 14