Question

How are the properties of exponents used to rewrite expressions?

Answer 1

In the explanation!

Explanation:

Generally speaking and considering any number OR expression: exponents are just quantities denoting the power of the number or expression to be raised to. There is a number of exponent properties available in the mathematical context, which are outlined below. Exponents basically help us to simplify computations and/or expressions during any type of calculations.

Properties Of Exponents:

Let us consider two numbers say
`a` and
`b` where
`a\\eq 0` and
`b\\eq 0` (i.e. Not Zero). We have the following rules:

• Product Rule:
  `a^x*a^y=a^(x+y)` Example:
  `2^2*2^3=2^5`

• Product Rule of Power:
  `(a^x)^y=a^(xy)` Example:
  `(2^2)^3=2^(6)`

• Quotient Rule:
  `(a^x)/(a^y)=a^(x-y)` Example:
  `(2^3)/(2^2)=2^1`

• Fraction Rule of Power:
  `((a)/(b))^x=(a^x)/(b^x)` Example:
  `((2)/(3) ^2)=(2^2)/(3^2)`

• Fractional Exponent:
  `a^{(x)/(y) }=\sqrt[y]{a^x}` Example:
  `2^{(2)/(3) }=\sqrt[3]{2^2}`

• Zero Exponent:
  `a^0 =1` Example:
  `2^0=1`

• Negative Exponent:
  `a^(-x)=(1)/(a^x)` Example:
  `2^(-3)=(1)/(2^3)`

These rules are applicable to all numbers and expressions in mathematics (including cases of ln, log, exponential 'e' and many others. They can be applied to either actual quantities (as shown above) or just varialbles. For example the Negative Exponent rule could also help us in a case such as:

`2^(-x)=(1)/(2^x)` and so on and so forth.