# Multiplying Exponents

To understand multiplication of exponents, let's start from a simple example.

Example. Suppose, we want to find (2^3)^4.

We already learned about positive integer exponets, so we can rewrite outer exponent: (2^3)^4=2^3*2^3*2^3*2^3.

Now, using addition of exponents, we have that (2^3)^4=2^3*2^3*2^3*2^3=2^(3+3+3+3)=2^(3*4)=2^12.

Let's see what have we done. We rewrote outer exponent, and then applied the rule for adding exponents.

But notice, that we added 3 four times, In other words we multiplied 3 by 4. Note, that 3*4=12.

It appears, that this rule works not only for positive integer exponents, it works for any exponent.

Rule for subtracting exponents: huge color(purple)((a^m)^n=a^(m*n)).

Note. Since m*n=n*m, then (a^m)^n=(a^n)^m.

Example 2. Find (2^3)^(-15).

It doesn't matter, that exponent is negative.

Just proceed as always: (2^3)^(-15)=2^(3*(-15))=2^(-45)=1/2^45.

Even when exponents are fractional, we use the same rule!

Example 3. Find (3^(1/4))^(2).

(3^(1/4))^2=3^(1/4*2)=3^(1/2)=sqrt(3).

We can handle radicals, also, because radicals can be rewritten with the help of exponent.

Example 4. Rewrite, using positive exponent: (root(7)(1/3^2))^5.

First we rewrite number, using exponents and then apply the rule:

(root(7)(1/3^2))^5=(root(7)(3^(-2)))^5=(3^(-2/7))^5=3^(-2/7*5)=3^(-10/7)=1/3^(10/7).

Now, it is time to exercise.

Exercise 1. Find (3^5)^2.

Answer: 3^10.

Exercise 2. Find (5^5)^(-2).

Answer: 5^(-10)=1/5^10.

Exercise 3. Find (4^(3/5))^5.

Answer: 4^3=64.

Exercise 4. Find (3^2)^(-1/5).

Answer: 3^(-2/5)=1/root(5)(1/9).

Exercise 5. Find (root(7)(1/27))^2.

Answer: (root(7)(3^(-3)))^2=(3^(-3/7))^2=3^(-6/7)=1/3^(6/7).