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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.