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.