what is i to the power of 100
Exponents are important in the financial earth, in scientific notation, and in the fields of epidemiology and public health. So what are they, and how do they work?
Exponents are written like \(3^two\) or \(x^3\).
Only what happens when y'all raise a number to the \(0\) power like this?
\[10^0 = \text{?}\]
This article volition go over
- the basics of exponents,
- what they mean, and
- it will prove that \(x^0\) equals \(1\) using negative exponents
All I'chiliad assuming is that you accept an understanding of multiplication and sectionalisation.
Exponents are made up of a base of operations and exponent (or power)
First, permit'due south start with the parts of an exponent.
There are two parts to an exponent:
- the base
- the exponent or power
At the beginning, nosotros had an exponent \(three^2\). The "iii" here is the base, while the "2" is the exponent or power.
We read this as
Three is raised to the power of two.
or
Three to the power of ii.
More generally, exponents are written as \(a^b\), where \(a\) and \(b\) can be any pair of numbers.
Exponents are multiplication for the "lazy"
Now that we accept some agreement of how to talk most exponents, how exercise we observe what number it equals?
Using our example from above, we can write out and expand "iii to the power of two" equally
\[3^2 = three \times 3 = 9\]
The left-most number in the exponent is the number we are multiplying over and over again. That is why y'all are seeing multiple three's. The right-most number in the exponent is the number of multiplications we exercise. So for our example, the number 3 (the base of operations) is multiplied two times (the exponent).
Some more examples of exponents are:
\[10^3 = x \times x \times 10 = 1000\]
\[two^{x} = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 1024 \]
More than generally, we tin write these exponents as
\[\textcolor{orange}{b}^\textcolor{blueish}{n} = \underbrace{\textcolor{orange}{b} \times \dots \times \textcolor{orange}{b}}_{\textcolor{blue}{due north} \textrm{ times}}\]
where, the \(\textcolor{orange}{\text{letter ``b'' is the base}}\) we are multiplying over and over again and the \(\textcolor{blue}{\text{letter ``n'' is power}}\) or \(\textcolor{bluish}{\text{exponent}}\), 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 over once more using multiplication signs.
In sum, exponents help brand writing these long multiplications more efficient.
Numbers to the power of naught are equal to one
The previous examples prove powers of greater than one, but what happens when it is nix?
The quick reply is that any number, \(b\), to the power of zilch is equal to i.
\[b^0 = ane\]
Based on our previous definitions, nosotros just need zero of the base of operations value. Here, let's accept our base number be 10.
\[10^0 = ? = one\]
But what does a "null" number of base numbers hateful? Why does this happen?
We can figure this out by dividing multiple times to decrease the power value until we get to zero.
Let's start with
\[10^3 = ten \times 10 \times 10 = 1000\]
To decrease the powers, we need to briefly understand the concepts of
- combining exponents
- powers of one
In our quest to decrease the exponent from \(10^3\) ("10 to the third power") to \(10^0\) ("10 to the zeroth power"), we will keep on doing the opposite of multiplying, which is dividing.
\[\frac{ten^3}{10} = \frac{ten \times ten \times 10}{10} = \frac{1000}{ten} = 100\]
The right-about parts of this will probably make sense. But how practise we write exponents when nosotros have \(10^3\) divided by \(10\)?
How powers of one work
First, whatever \(\textcolor{orange}{\text{exponents with powers of one}}\) are equal to just \(\textcolor{blue}{\text{the base number}}\).
\[\textcolor{orangish}{b^1} = \textcolor{blue}{b}\]
There is simply i value beingness "multiplied" so we are getting the value itself.
We need this "ability of 1" definition so we can rewrite the fraction with exponents.
\[\frac{10^three}{10} = \frac{10^3}{ten^one}\]
How to decrease exponents to zero
Every bit a reminder, one way to effigy out how \(10^0\) is equal to ane is to keep on dividing by ten until we get to an exponent of null.
We know from the right side of the equation above we should get 100 from \(\frac{10^three}{ten^1}\).
\[ \frac{ten^3}{x} = \frac{10^3}{10^1} = \frac{ten \times x \times 10}{ten^1} \]
Earlier we finish dividing by one 10, we can multiply the superlative and bottom by 1 as placeholders when nosotros cancel numbers out.
\[ \frac{10 \times x \times 10}{ten^ane} = \frac{10 \times x \times 10 \times i}{10^ane \times 1} = \frac{10 \times x \times \cancel{10} \times 1}{\cancel{10^1} \times one} = \frac{10 \times 10 \times one}{1}\]
From this, nosotros can see we get 100 again.
\[ \frac{10 \times 10 \times one}{1} = \frac{x \times x}{1} = \frac{10^2}{1} = \frac{100}{one} \]
We tin divide by 10 two more times to finally get to \(10^0\).
\[ \frac{ten^2 \times 1}{x \times x \times one} = \frac{\cancel{ten} \times \cancel{x} \times 1}{\cancel{10} \times \abolish{ten} \times i} = \frac{10^0 \times 1}{1} = \frac{ane}{1} = 1 \]
Considering we divided by two 10's when we only had 2 10'due south in the meridian of the fraction, we have naught tens in the meridian. Having zero tens pretty much means we get \(10^0\).
How negative exponents work
Now, the \(ten^0\) kind of comes out of nowhere, and then allow's explore this some more using "negative exponents".
More generally, this repetitive dividing by the same base of operations is the aforementioned equally multiplying by "negative exponents".
A negative exponent is a way to rewrite sectionalization.
\[ \frac{one}{\textcolor{royal}{b^n}}= \textcolor{green}{b^{-northward}}\]
A \(\textcolor{green}{\text{negative exponent}}\) can exist re-written as a fraction with the denominator (or the lesser of a fraction) with the \(\textcolor{imperial}{\text{same exponent only with a positive power}}\) (the left side of this equation).
Now, using negative exponents, we tin show the previous sectionalisation in some other way.
\[ \frac{10^2 \times 1}{10 \times 10 \times i} = \frac{10^ii}{10^ii} = 10^2 \times \frac{ane}{10^two} = ten^2 \times 10^{-2} \]
Note, i rule of exponents is that when you multiply exponents with the same base number (call back, our base number hither is 10), you tin add the exponents.
\[ x^2 \times x^{-2} = 10^{2 + (-two)} = 10^{2 - 2} = 10^{0} \]
Putting it together
Knowing this, we tin can combine each of these equations in a higher place to summarize our outcome.
\[ \textcolor{purple}{\frac{10^2}{10^two}} = x^2 \times 10^{-2} = 10^{2 + (-2)} = 10^{two - 2} = \textcolor{bluish}{10^{0}} \textcolor{orange}{= 1} \]
We know that \(\textcolor{purple}{\text{dividing a number past itself}}\) volition \(\textcolor{orangish}{\text{equal to one}}\). And nosotros've shown that \(\textcolor{purple}{\text{dividing a number by itself}}\) likewise equals \(\textcolor{blue}{\text{ten to the nothing power}}\). Math says that things that are equal to the same thing are as well equal to each other.
Thus, \(\textcolor{bluish}{\text{ten to the zero power}}\) is \(\textcolor{orangish}{\text{equal to ane}}\). This exercise above generalizes to any base of operations number, so whatever number to the power of nix is equal to one.
In summary
Exponents are user-friendly ways to do repetitive multiplication.
Generally, exponents follow this blueprint below, with some \(\textcolor{orange}{\text{base number}}\) existence multiplied over and over once again \(\textcolor{blue}{\text{``n'' number of times}}\).
\[\textcolor{orange}{b}^\textcolor{blue}{northward} = \underbrace{\textcolor{orange}{b} \times \dots \times \textcolor{orange}{b}}_{\textcolor{blueish}{n} \textrm{ times}}\]
Using negative exponents, we can take what we know from multiplication and division (like for the fraction 10 over 10,\(\frac{10}{10}\)) to show that \(b^0\) is equal to i for whatsoever number \(b\) (like \(10^0 = 1\)).
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Cheers for reading!
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Source: https://www.freecodecamp.org/news/10-to-the-power-of-0-the-zero-exponent-rule-and-the-power-of-zero-explained/
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