# Properties of Exponents (Rules)

Properties (rules) of exponents:

• Zero power: a^0=1, a!=0
• Zero base: 0^a=0, a!=0
• 0^0 is undefined
• 1^a=1
• Negative exponent: a^(-b)=1/a^b, b!=0
• Nth root: a^(1/n)=root(n)(a), n!=0
• Addition of exponents: a^m*a^n=a^(m+n)
• Subtraction of exponents: (a^m)/(a^n)=a^(m-n), a!=0
• Multiplication of exponents: (a^m)^n=a^(m*n)=(a^n)^m
• Division of exponents: root(n)(a^m)=a^(m/n), n!=0
• root(m)(a^m)=a, if m is odd
• root(m)(a^m)=|a|, if m is even
• root(n)(a^m)=(root(n)(a))^m (just pay attention to signs and check, whether number exists)
• Power of a product: a^n*b^n=(ab)^n
• Power of a quotient: (a^n)/(b^n)=(a/b)^n, b!=0

We already covered all rules earlier, except last two.

To understand last two properties, consider the following example.

Example. Find 2^3*4^3.

Let's rewrite numbers: (color(red)(2))^3*(color(green)(4))^3=color(red)(2*2*2)*color(green)(4*4*4).

Now, regroup: color(red)(2*2*2)*(color(green)(4*4*4))=(color(red)(2)*color(green)(4))*(color(red)(2)*color(green)(4))*(color(red)(2)*color(green)(4))=(2*4)^3.

Note, that on the last step, we wrapped the product, using exponent.

This property is valid for any exponent, so:

Power of a product: huge color(purple)(a^n*b^n=(a*b)^n).

Similarly, it can be shown that a^n/b^n=(a/b)^n.

Power of a quotient: huge color(purple)(a^n/b^n=(a/b)^n), b!=0.

We can combine above rules to simplify more complex examples.

Example 2. Find 6^4/3^4.

Using power of a quotient rule, we can write, that 6^4/3^4=(6/3)^4=2^4=16.

Now, let's see how combination of rules works.

Example 3. Rewrite, using positive exponents: root(5)(3^2*3^5).

We first apply rule for adding exponents: root(5)(3^2*3^5)=root(5)(3^(2+5))=root(5)(3^7).

Now, apply rule for dividing exponents: root(5)(3^7)=3^(7/5).

So, root(5)(3^2*3^5)=root(5)(3^7).

Finally, let's see how to apply more than two rules.

Example 4. Rewrite, using positive exponents ((root(5)(3)*root(5)(4))/12^3)^3.

First, we rewrite using exponents: ((root(5)(3)*root(5)(4))/12^3)^3=((3^(1/5)*4^(1/5))/12^3)^3.

Now, apply power of a product rule: ((3^(1/5)*4^(1/5))/12^3)^3=(((3*4)^(1/5))/12^3)^3=((12^(1/5))/12^3)^3.

Next, use rule for subtracting exponents: ((12^(1/5))/12^3)^3=(12^(1/5-3))^3=(12^(-14/5))^3.

Next, apply rule for multiplying exponents: (12^(-14/5))^3=12^(-14/5*3)=12^(-42/5).

Finally, apply negative exponent rule: 12^(-42/5)=1/12^(42/5).

Answer: 1/12^(42/5).

Now, practice a little.

Exercise 1. Rewrite, using positive exponents: root(4)(2*2^5).

Answer: 2^(3/2).

Exercise 2. Rewrite, using positive exponents: (2^3/4^5*root(5)(5^7))^0.

Answer: 1. Hint: as long as base is non-zero, raising to zero power gives 1.

Exercise 3. Rewrite, using positive exponents: root(5)(root(4)(2^3*6^3*12^5)).

Answer: 12^(2/5).

Exercise 4. Simplify: ((2^3)/(6^3)*3^5)^7.

Answer: 3^14.

Exercise 5. Rewrite, using positive exponents: root(7)((root(5)((-2)^7)/(-2^3)*(-2)^7)^2).

Answer: 2^(54/35). Hint. pay attention to signs: ((-2)^(27/5))^2=2^(54/5). Minus vanishes, because we square.