what is any number to the power of 1
Exponents are of import in the financial earth, in scientific notation, and in the fields of epidemiology and public wellness. So what are they, and how do they piece of work?
Exponents are written like \(3^2\) or \(10^3\).
Merely what happens when you raise a number to the \(0\) power like this?
\[10^0 = \text{?}\]
This commodity will go over
- the nuts of exponents,
- what they mean, and
- it will show that \(10^0\) equals \(1\) using negative exponents
All I'thousand assuming is that you lot have an understanding of multiplication and segmentation.
Exponents are made up of a base and exponent (or ability)
Beginning, let's start with the parts of an exponent.
In that location are ii parts to an exponent:
- the base
- the exponent or power
At the beginning, we had an exponent \(3^two\). The "3" here is the base of operations, while the "2" is the exponent or ability.
We read this equally
3 is raised to the power of two.
or
Three to the power of 2.
More generally, exponents are written equally \(a^b\), where \(a\) and \(b\) can be any pair of numbers.
Exponents are multiplication for the "lazy"
Now that nosotros have some understanding of how to talk about exponents, how do we detect what number it equals?
Using our case from in a higher place, we can write out and expand "three to the power of two" as
\[3^2 = three \times three = 9\]
The left-most number in the exponent is the number nosotros are multiplying over and over once more. That is why you are seeing multiple 3's. The right-most number in the exponent is the number of multiplications we exercise. And so for our case, the number three (the base) is multiplied 2 times (the exponent).
Some more examples of exponents are:
\[10^three = 10 \times x \times 10 = thou\]
\[2^{10} = ii \times 2 \times 2 \times 2 \times 2 \times 2 \times ii \times ii \times two \times 2 = 1024 \]
More generally, nosotros can write these exponents as
\[\textcolor{orange}{b}^\textcolor{blue}{n} = \underbrace{\textcolor{orange}{b} \times \dots \times \textcolor{orange}{b}}_{\textcolor{bluish}{due north} \textrm{ times}}\]
where, the \(\textcolor{orange}{\text{letter ``b'' is the base}}\) we are multiplying over and over over again and the \(\textcolor{blue}{\text{letter ``n'' is ability}}\) or \(\textcolor{blue}{\text{exponent}}\), which is the number of times we are multiplying the base past itself.
For these examples above, the exponent values are relatively small. But you tin can imagine if the powers are very large, it becomes redundant to keep writing the numbers over and once more using multiplication signs.
In sum, exponents help make writing these long multiplications more efficient.
Numbers to the power of zero are equal to i
The previous examples prove powers of greater than one, simply what happens when it is nix?
The quick answer is that any number, \(b\), to the ability of zero is equal to one.
\[b^0 = 1\]
Based on our previous definitions, we just need cipher of the base value. Here, let's have our base number be 10.
\[10^0 = ? = 1\]
But what does a "nada" number of base numbers mean? Why does this happen?
We can figure this out by dividing multiple times to decrease the power value until we get to zero.
Permit's starting time with
\[x^iii = x \times 10 \times ten = yard\]
To decrease the powers, we need to briefly sympathize the concepts of
- combining exponents
- powers of one
In our quest to decrease the exponent from \(10^3\) ("ten to the third power") to \(10^0\) ("ten to the zeroth ability"), we volition keep on doing the opposite of multiplying, which is dividing.
\[\frac{10^iii}{10} = \frac{10 \times ten \times 10}{10} = \frac{1000}{10} = 100\]
The right-most parts of this volition probably make sense. But how do we write exponents when we have \(x^3\) divided past \(10\)?
How powers of i piece of work
Start, whatever \(\textcolor{orange}{\text{exponents with powers of one}}\) are equal to only \(\textcolor{blue}{\text{the base number}}\).
\[\textcolor{orange}{b^1} = \textcolor{blue}{b}\]
There is only one value being "multiplied" so nosotros are getting the value itself.
We need this "power of ane" definition so nosotros can rewrite the fraction with exponents.
\[\frac{ten^3}{x} = \frac{x^iii}{10^1}\]
How to subtract exponents to zero
As a reminder, 1 fashion to effigy out how \(10^0\) is equal to 1 is to keep on dividing by 10 until we get to an exponent of nada.
Nosotros know from the correct side of the equation above we should get 100 from \(\frac{10^iii}{ten^1}\).
\[ \frac{10^3}{10} = \frac{x^3}{10^ane} = \frac{ten \times 10 \times ten}{10^one} \]
Before we finish dividing by one ten, we can multiply the pinnacle and bottom by one equally placeholders when nosotros cancel numbers out.
\[ \frac{10 \times 10 \times 10}{10^one} = \frac{10 \times 10 \times ten \times one}{10^one \times 1} = \frac{10 \times 10 \times \cancel{10} \times 1}{\cancel{x^1} \times 1} = \frac{10 \times ten \times 1}{1}\]
From this, we tin see we go 100 over again.
\[ \frac{10 \times 10 \times 1}{i} = \frac{10 \times 10}{1} = \frac{10^2}{1} = \frac{100}{ane} \]
We can divide past x two more times to finally become to \(10^0\).
\[ \frac{10^2 \times ane}{10 \times 10 \times ane} = \frac{\cancel{x} \times \abolish{10} \times i}{\cancel{10} \times \cancel{ten} \times 1} = \frac{10^0 \times i}{one} = \frac{1}{1} = one \]
Because we divided by 2 10'due south when we just had two 10'south in the height of the fraction, we have goose egg tens in the superlative. Having zero tens pretty much ways we get \(ten^0\).
How negative exponents piece of work
Now, the \(ten^0\) kind of comes out of nowhere, then let'southward explore this some more using "negative exponents".
More more often than not, this repetitive dividing by the same base is the same equally multiplying past "negative exponents".
A negative exponent is a way to rewrite division.
\[ \frac{ane}{\textcolor{purple}{b^n}}= \textcolor{dark-green}{b^{-n}}\]
A \(\textcolor{greenish}{\text{negative exponent}}\) can be re-written equally a fraction with the denominator (or the bottom of a fraction) with the \(\textcolor{purple}{\text{same exponent but with a positive power}}\) (the left side of this equation).
Now, using negative exponents, we can show the previous division in some other way.
\[ \frac{10^2 \times 1}{10 \times 10 \times 1} = \frac{ten^2}{10^2} = ten^2 \times \frac{1}{10^2} = x^2 \times 10^{-2} \]
Note, one rule of exponents is that when you multiply exponents with the same base number (call back, our base number hither is x), you can add the exponents.
\[ 10^2 \times 10^{-ii} = 10^{2 + (-ii)} = 10^{2 - 2} = ten^{0} \]
Putting it together
Knowing this, we can combine each of these equations above to summarize our result.
\[ \textcolor{purple}{\frac{ten^2}{ten^2}} = 10^2 \times x^{-two} = ten^{ii + (-2)} = 10^{two - 2} = \textcolor{blue}{10^{0}} \textcolor{orange}{= 1} \]
We know that \(\textcolor{royal}{\text{dividing a number by itself}}\) will \(\textcolor{orange}{\text{equal to 1}}\). And we've shown that \(\textcolor{purple}{\text{dividing a number by itself}}\) as well equals \(\textcolor{bluish}{\text{ten to the zero power}}\). Math says that things that are equal to the aforementioned thing are also equal to each other.
Thus, \(\textcolor{blue}{\text{x to the zero power}}\) is \(\textcolor{orange}{\text{equal to i}}\). This practise above generalizes to any base number, so any number to the power of zilch is equal to one.
In summary
Exponents are convenient ways to practise repetitive multiplication.
Generally, exponents follow this design below, with some \(\textcolor{orangish}{\text{base number}}\) beingness multiplied over and over once more \(\textcolor{bluish}{\text{``due north'' number of times}}\).
\[\textcolor{orange}{b}^\textcolor{blue}{n} = \underbrace{\textcolor{orange}{b} \times \dots \times \textcolor{orange}{b}}_{\textcolor{bluish}{due north} \textrm{ times}}\]
Using negative exponents, nosotros can have what we know from multiplication and sectionalisation (like for the fraction x over ten,\(\frac{10}{10}\)) to evidence that \(b^0\) is equal to one for any number \(b\) (similar \(10^0 = 1\)).
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