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)`.