What is 10 raised to the power of 0?
Exponents are of import in the financial world, in scientific note, and in the fields of epidemiology and public health. Then what are they, and how practise they work?
Exponents are written similar \(three^ii\) or \(ten^three\).
But what happens when you heighten a number to the \(0\) power similar this?
\[ten^0 = \text{?}\]
This article will go over
- the basics of exponents,
- what they hateful, and
- information technology will prove that \(10^0\) equals \(ane\) using negative exponents
All I'1000 assuming is that yous have an understanding of multiplication and division.
Exponents are fabricated upward of a base and exponent (or power)
Kickoff, permit's start with the parts of an exponent.
There are two parts to an exponent:
- the base
- the exponent or power
At the beginning, we had an exponent \(3^2\). The "three" 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 two.
More by and large, 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 have some understanding of how to talk almost exponents, how do we notice what number it equals?
Using our example from above, nosotros can write out and expand "3 to the power of two" as
\[3^ii = iii \times three = 9\]
The left-virtually number in the exponent is the number we are multiplying over and over once again. That is why you are seeing multiple 3'south. The correct-about number in the exponent is the number of multiplications we do. And then for our case, the number iii (the base) is multiplied ii times (the exponent).
Some more examples of exponents are:
\[10^3 = 10 \times x \times 10 = 1000\]
\[2^{10} = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times ii \times 2 = 1024 \]
More generally, we tin write these exponents every bit
\[\textcolor{orange}{b}^\textcolor{blue}{n} = \underbrace{\textcolor{orangish}{b} \times \dots \times \textcolor{orange}{b}}_{\textcolor{blue}{n} \textrm{ times}}\]
where, the \(\textcolor{orange}{\text{letter ``b'' is the base}}\) nosotros are multiplying over and once again and the \(\textcolor{blue}{\text{letter of the alphabet ``n'' is power}}\) or \(\textcolor{blue}{\text{exponent}}\), which is the number of times we are multiplying the base by itself.
For these examples in a higher place, the exponent values are relatively small. Simply you can imagine if the powers are very large, it becomes redundant to keep writing the numbers over and over once again using multiplication signs.
In sum, exponents help make writing these long multiplications more efficient.
Numbers to the power of zero are equal to one
The previous examples testify powers of greater than i, simply what happens when it is nada?
The quick answer is that any number, \(b\), to the power of zippo is equal to ane.
\[b^0 = 1\]
Based on our previous definitions, we just need aught of the base value. Hither, permit's accept our base number be 10.
\[10^0 = ? = 1\]
Just what does a "zero" number of base numbers mean? Why does this happen?
Nosotros can figure this out by dividing multiple times to decrease the power value until we get to goose egg.
Permit's showtime with
\[10^3 = 10 \times 10 \times 10 = thousand\]
To decrease the powers, we demand to briefly empathise the concepts of
- combining exponents
- powers of i
In our quest to subtract the exponent from \(x^3\) ("ten to the 3rd ability") to \(10^0\) ("ten to the zeroth power"), we will go on on doing the opposite of multiplying, which is dividing.
\[\frac{ten^3}{ten} = \frac{x \times x \times ten}{10} = \frac{chiliad}{10} = 100\]
The right-most parts of this will probably make sense. But how do we write exponents when we have \(x^3\) divided by \(x\)?
How powers of ane piece of work
Get-go, any \(\textcolor{orange}{\text{exponents with powers of one}}\) are equal to just \(\textcolor{bluish}{\text{the base number}}\).
\[\textcolor{orangish}{b^one} = \textcolor{blue}{b}\]
There is only one value beingness "multiplied" and then we are getting the value itself.
We need this "power of one" definition then nosotros tin rewrite the fraction with exponents.
\[\frac{x^iii}{10} = \frac{10^iii}{ten^1}\]
How to decrease exponents to zero
Every bit a reminder, 1 way to effigy out how \(10^0\) is equal to i is to keep on dividing past ten until we get to an exponent of zero.
Nosotros know from the right side of the equation higher up we should become 100 from \(\frac{ten^3}{x^1}\).
\[ \frac{10^3}{10} = \frac{10^3}{10^one} = \frac{10 \times 10 \times 10}{10^1} \]
Before we finish dividing past ane 10, nosotros tin multiply the height and lesser by 1 as placeholders when we cancel numbers out.
\[ \frac{10 \times 10 \times 10}{ten^1} = \frac{x \times 10 \times x \times one}{ten^1 \times 1} = \frac{x \times 10 \times \cancel{10} \times 1}{\cancel{10^1} \times ane} = \frac{10 \times 10 \times 1}{1}\]
From this, we can come across nosotros get 100 again.
\[ \frac{10 \times ten \times ane}{one} = \frac{x \times x}{i} = \frac{10^two}{1} = \frac{100}{one} \]
We tin can divide by 10 two more times to finally get to \(10^0\).
\[ \frac{10^2 \times one}{10 \times 10 \times 1} = \frac{\cancel{10} \times \cancel{ten} \times 1}{\cancel{10} \times \cancel{10} \times 1} = \frac{10^0 \times one}{1} = \frac{1}{1} = 1 \]
Because nosotros divided by two 10's when nosotros only had two ten's in the top of the fraction, we take zero tens in the superlative. Having nil tens pretty much means we get \(10^0\).
How negative exponents work
Now, the \(10^0\) kind of comes out of nowhere, then let's explore this some more than using "negative exponents".
More generally, this repetitive dividing by the aforementioned base is the same as multiplying by "negative exponents".
A negative exponent is a way to rewrite sectionalization.
\[ \frac{ane}{\textcolor{purple}{b^northward}}= \textcolor{green}{b^{-n}}\]
A \(\textcolor{dark-green}{\text{negative exponent}}\) can be re-written as a fraction with the denominator (or the bottom of a fraction) with the \(\textcolor{regal}{\text{aforementioned exponent just with a positive power}}\) (the left side of this equation).
At present, using negative exponents, we can show the previous division in another mode.
\[ \frac{x^2 \times 1}{10 \times 10 \times i} = \frac{x^two}{ten^2} = 10^2 \times \frac{1}{10^two} = ten^two \times x^{-2} \]
Note, one rule of exponents is that when you lot multiply exponents with the same base number (remember, our base of operations number hither is 10), you can add the exponents.
\[ 10^two \times 10^{-ii} = x^{2 + (-2)} = 10^{2 - 2} = 10^{0} \]
Putting information technology together
Knowing this, we can combine each of these equations above to summarize our result.
\[ \textcolor{majestic}{\frac{10^2}{10^ii}} = x^2 \times 10^{-2} = 10^{2 + (-2)} = x^{2 - two} = \textcolor{blue}{10^{0}} \textcolor{orangish}{= 1} \]
We know that \(\textcolor{purple}{\text{dividing a number by itself}}\) will \(\textcolor{orange}{\text{equal to ane}}\). And we've shown that \(\textcolor{purple}{\text{dividing a number by itself}}\) also equals \(\textcolor{bluish}{\text{ten to the zero ability}}\). Math says that things that are equal to the aforementioned affair are also equal to each other.
Thus, \(\textcolor{blueish}{\text{ten to the zero ability}}\) is \(\textcolor{orange}{\text{equal to one}}\). This exercise above generalizes to any base number, so any number to the power of nothing is equal to ane.
In summary
Exponents are convenient ways to do repetitive multiplication.
Generally, exponents follow this pattern below, with some \(\textcolor{orange}{\text{base of operations number}}\) existence multiplied over and over again \(\textcolor{bluish}{\text{``n'' number of times}}\).
\[\textcolor{orange}{b}^\textcolor{blue}{north} = \underbrace{\textcolor{orange}{b} \times \dots \times \textcolor{orange}{b}}_{\textcolor{blue}{n} \textrm{ times}}\]
Using negative exponents, we can take what we know from multiplication and sectionalisation (like for the fraction 10 over ten,\(\frac{10}{10}\)) to show that \(b^0\) is equal to i for any number \(b\) (like \(10^0 = 1\)).
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Thanks for reading!
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