\[x^{1}= x\]
Examples
\[14^{1}= 14\]
\[(-9)^{1}= -9\]
\[345^{1}= 345\]
Why It Works
When you simplify powers with positive exponents, the exponent tells you how many times to multiply the base by itself.
For example, \(2^{6}\) means that you need to multiply 2 by itself 6 times.
\[2^{6}= 2\times 2\times 2\times 2\times 2\times 2 = 64\]
When you have a exponent of 1, it means that you need to “multiply” the base by itself just once.
Technically, it isn’t actually multiplying because you need at least two numbers to do “multiplication”, but we don’t have to be technical 😉
\[2^{1}= 2\]
The "Invisible" Exponent of 1
Raising a number or variable to an exponent of 1 doesn’t change the value of the expression at all.
So, when it’s helpful, you can imagine that there’s an “invisible” exponent of 1 for any number or variable that doesn’t already have an exponent.
\[12=12^{1}\]
\[x=x^{1}\]
\[4y=4^{1}y^{1}\]
This is similar to the invisible coefficient of 1 we often use for terms that don’t have coefficients.
The invisible exponent of 1 is very helpful when you’re using exponent rules or working with polynomials.
For example, if you were trying to use the exponent rules, it may be helpful to think of each of these expressions…
You may want to think of \(x^{5}x\) as \(x^{5}x^{1}\) before you use the Product Rule for Exponents.
It could be helpful to think of \((4xy^{2})^{3}\) as \((4^{1}x^{1}y^{2})^{3}\) before you use the Distributive Property for Exponents.
If you were asked to find the degree of this polynomial \(xy^{3}+x^{2}y+xy\), it may be helpful to think of it like this: \(x^{1}y^{3}+x^{2}y^{1}+x^{1}y^{1}\)
Identity Property of Exponents
The identity property of addition says that any number plus 0 will equal itself.
The identity property of multiplication says that any number times 1 will equal itself.
Add
\[a+0=0\]
Multiply
\[a(1)=a\]
Exponent
\[a^1=a\]
A power raised to an exponent of 1 could be called the “Identity Property of Exponents”. The exponent of 1 would be the “Exponential Identity” of that property.
As far as I know, this terminology doesn’t exist in any textbooks, but I think it makes sense, so I’m making it up 🙂
I like thinking about it this way because it shows how many similarities there are between all of the math operations.
Exponential Identity = 1
Multiplicative Identity = 1
Additive Identity = 0
