The term that you are looking for, the result when a base is "raised to a power" is power … and it is incorrect to say "a base raised to a power". We should say "a base raised to an exponent, which is a power". The term power is one of the most widely misused in all of mathematics. Let me explain.
Let’s say that Tom has 5 children. They are, in order starting the the first: A, B, C, D and E. The third child of Tom is C.
Tom is the parent of this list or sequence of children. We will then refer to Tom as the "root" or the "base".
Next, the multiples of 2 are: 2, 4, 6, 8, 10, …
This sequence is based on 2 (it starts at 2 and we add 2 to get each successive number), so 2 is the base.
The third multiple of 2 is 6.
2 + 2 + 2 = 6.
We can condense this as 3x2 = 6 (3 copies of 2, added together equals 6).
6 is the sum, or in the case of repeated addition, the multiple.
Of whom? Of 2.
Which multiple? The third one.
The powers of 2 are: 2, 4, 8, 16, 32, …
The base of this sequence is 2 because we start at 2 and multiply by 2 to get each successive number.
The third power of 2 is 8.
2 x 2 x 2 = 8.
We can condense this as 2^3 = 8 (3 copies of 2, multiplied together equals 8).
8 is the product, or in the case of repeated multiplication of a base number, the power.
Of whom? Of 2.
Which power? The third one.
- C is the third child of Tom.
- 6 is the third multiple of 2.
- 8 is the third power of 2.
So who is the third
- child? C … not 3
- multiple? 6 … not 3
- power? 8 … not 3! 3 is the *not the power, 3 is the exponent! 8 is the power!
In 2^3 = 8, 2 is the base, 3 is the exponent and 8 is the power. Specifically 8 is the third power of 2.
The term power is the most widespread and consistently grammatically misused word in all of mathematics. The power is NOT the exponent, ever. You cannot read 2^3 as "2 raised to the power 3" or "2 raised to the third power" or "2 to the third" (that sounds like 2^(1/3) which is the cube root of 2, not 2^3 = 8).
The third power of 2 is 8. So 8 is the power when 2 is the base and 3 is the exponent.
Read 2^3 as "2 raised to the exponent of 3", "2 to the 3" (though I prefer that learners emphasise that 3 is the exponent) or, in this specific example, "2 cubed".
It may seem like splitting hairs to distinguish between "exponent" and "power" but it is crucial to understanding exponential and logarithmic functions.
If y = 2^x, then for the exponential base 2 function, the input x is an exponent and the output is a power. If we cannot distinguish inputs from outputs, deep and authentic understanding cannot be achieved.
It would be like saying in 3x5=15 that 15 is the product. And so are 3 and 5. Which is false, because product is the output of multiplying two (or more) input factors together. It would be like saying that area of a rectangle = length x width: A = l x w, so the output A is area … and so are the inputs l and w also area!
So, b^x = p says that b is the base, x is the exponent and p is the power … regardless of what you read on the internet.