What are exponents? Give some properties and facts for the same.
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An exponent refers to the number of times a number is multiplied by itself. For example, 2 to the 3rd (written like this: 23) means: 2 x 2 x 2 = 8. 23 is not the same as 2 x 3 = 6. Remember that a number raised to the power of 1 is itself.
Exponentiation is a mathematical operation, written as bⁿ, involving two numbers, the base b and the exponent or power n.
Properties of exponents
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
c=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−−√2=x12
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