UCSB Science Line
The following base, exponent, root, and logarithm notation can be entered in Show My Exponent and subscript of a variable, exponent and subscript button . Adding and subtracting with scientific notation may require more care, because Remember that to be like terms, two expressions must have exactly the same base numbers to exactly the same powers. We work around this by using our exponent property bm · bn = b (m+n) to rewrite Date last modified: July 22, Question Date: Answer 1: This is an excellent question! There are lots of different ways to think about it, but here's one: let's go back and think about One rule for exponents is that exponents add when you have the same base.
If it ever sounds arbitrary then hound your teacher.Math - How to multiply and divide exponent with different base (Law VI and VII) -English
If your teacher can't give you compelling reasons why something is true, hound us or hound Google. Okay, enough, onto your question: Mathematics was initially developed to describe relationships between everyday quantities generally whole numbers so the best way to think about powers like ab 'a' raised to the 'b' power is that the answer represents the number of ways you can arrange sets of 'b' numbers from 1 to 'a'.
For example, 23 is 8. There are 8 ways to write sets of 3 numbers where each number can be either 1 or 2: So what does 30 represent?
Math Forum - Ask Dr. Math
It is the number of ways you can arrange the numbers 1,2, and 3 into lists containing none of them! How many ways are there to place a penny, a nickel, and a quarter on the table such that no coins are on the table?
There are other reasons why a0 has to be 1 - for example, you may have heard the power rule: I know this sounds a little fishy since we started with a rule I could have just made up which is why I gave the other reason firstbut these formulas are all consistent and there is never any magic step, I promise! Any number to the zero power always gives one. One rule for exponents is that exponents add when you have the same base.
So if you have a number, x, and exponents, a and b, then: So now we've shown that: If you had trouble understanding it all with variables, let's look at it again,but this time as an example with numbers: I don't see how you can raise something to a power, and get the power as the answer. The book's terminology is correct; but the English usage here is awkward and is often misinterpreted, not only by students but even by textbook authors and lexicographers. That, of course, means that you can find many authorities for a different view than mine!
In fact, my understanding of the terms may be a minority view; but I think it is correct.
Laws of Exponents
When we write 3 2 we say that 2 is the "base", 3 is the "exponent", and the whole thing is "a power of 2" in particular, "the third power of 2". The awkwardness comes from the fact that we call this expression "2 to the third power" or "2 raised to the power [of] 3". This SOUNDS as if we were saying that 3 was the "power" to which we raised 2, and as a result the word "power" is, as you point out, often used interchangeably with "exponent".
We are raising the number 2 TO a power, changing it from its original "weak" form to a more "powerful" form; in fact, we have raised it to its third power, the third level it can reach.
The power is the number it got to, not the number of steps it took to get there. I believe that the phrase started with "the third power of 2", which clearly names the result of the operation as a power; then moved to "2 raised to the third power", which means the same thing but emphasizes the operation of "raising" rather than the value; and then, when variable exponents were needed, had to be twisted around to "2 raised to the power of x" to avoid having to say "2 raised to the xth power".
And at that point, it started sounding as if x was a power, though even here you can still see a distinction, in that the power is "of x", that is, belonging to, or associated with, x, not x itself. When people say "2 raised to the power x", that distinction is lost.
2 Easy Ways to Add Exponents (with Pictures) - wikiHow
So I can't blame people for getting confused, and I have to recognize that the term "power" is very commonly used to mean "exponent". But I think it is useful to retain a word that refers to the whole expression just as we use "product" to refer to the result of multiplication ; and the only word available is "power"!
Can you think of an alternative, if we reserve "power" to mean the exponent?