List all the properties of exponents and powers.
Answers
Answer:
Formulas and properties of exponents
1. a0 = 1 (a ≠ 0) - zero exponent property
2. a1 = a - any number raised by the exponent 1 is the number itself
3. an · am = an + m - product of powers property
4. (an)m = anm - power of a power property
5. anbn = (ab)n - power of a product property
Answer:
In earlier chapters we introduced powers.
x3=x⋅x⋅x
There are a couple of operations you can do on powers and we will introduce them now.
We can multiply powers with the same base
x4⋅x2=(x⋅x⋅x⋅x)⋅(x⋅x)=x6
This is an example of the product of powers property tells us that when you multiply powers with the same base you just have to add the exponents.
xa⋅xb=xa+b
We can raise a power to a power
(x2)4=(x⋅x)⋅(x⋅x)⋅(x⋅x)⋅(x⋅x)=x8
This is called the power of a power property and says that to find a power of a power you just have to multiply the exponents.
When you raise a product to a power you raise each factor with a power
(xy)2=(xy)⋅(xy)=(x⋅x)⋅(y⋅y)=x2y2
This is called the power of a product property
(xy)a=xaya
As well as we could multiply powers we can divide powers.
x4x2=x⋅x⋅⧸x⋅⧸x⧸x⋅⧸x=x2
This is an example of the quotient of powers property and tells us that when you divide powers with the same base you just have to subtract the exponents.
xaxb=xa−b,x≠0
When you raise a quotient to a power you raise both the numerator and the denominator to the power.
(xy)2=xy⋅xy=x⋅xy⋅y=x2y2
This is called the power of a quotient power
(xy)a=xaya,y≠0
When you raise a number to a zero power you'll always get 1.
1=xaxa=xa−a=x0
x0=1,x≠0
Negative exponents are the reciprocals of the positive exponents.
x−a=1xa,x≠0
xa=1x−a,x≠0
The same properties of exponents apply for both positive and negative exponents.
In earlier chapters we talked about the square root as well. The square root of a number x is the same as x raised to the 0.5th power
x−−√=x−−√2=x1