1 power rules
10^💪
1.1 product rule
It is no surprise that
100 \cdot 1000 = 100000.
Using power notation, the same can be written as
10^2 \cdot 10^3 = 10^{2+3} = 10^5.
Now we’re getting at something. Generalizing the pattern above, we can write the one important rule we need, the product rule:
10^a \cdot 10^b = 10^{a+b}.
1.2 zero exponent rule
From the product rule above we can derive all sorts of other useful relationships. Let’s start by taking b=0 in the product rule:
10^a \cdot 10^0 = 10^{a+0} = 10^{a}.
The only way to make the product rule true when one of the exponents is zero is to conclude the following:
10^0 = 1
1.3 negative exponent rule
What happens if one of the exponents is negative? For example, let’s substitute b=-a in the product rule:
10^a \cdot 10^{-a} = 10^{a-a} = 10^0 = 1.
If we divide both sides of the equation above by 10^a we get
10^{-a} = \frac{1}{10^a}.
We’ve found that negative powers are related to reciprocals. (1/x is called the reciprocal of x)
1.4 quotient rule
Now that we know how to deal with negative exponents, we can deal with a power divided by another power:
10^a \div 10^b = 10^a \cdot \frac{1}{10^b} = 10^a \cdot 10^{-b} = 10^{a-b},
or in short
10^a \div 10^b = 10^{a-b}.
1.5 power of a power rule
What happens when we raise a power to another power?
(10^a)^b =\underbrace{10^a\cdot10^a\cdots10^a}_{b\text{ factors}} =10^{\overbrace{a+a+\cdots+a}^{b\text{ terms}}} =10^{ab}.
So, in short:
(10^a)^b = 10^{ab}.
1.6 fractional exponent rule
What should a fractional exponent mean? Start with 10^{1/2}. Using the power-of-a-power rule:
(10^{1/2})^2=10^{\frac{1}{2}\cdot 2} = 10.
We don’t really want the square of 10^{1/2}, so how to undo the squaring operation? We take the square root, \sqrt[2]{\,\,}. So we’ve learned that 10^{1/2}=\sqrt[2]{10}.
The same reasoning works for any fraction a/b (given that b is positive):
(10^{a/b})^b = 10^{\frac{a}{b}\cdot b} = 10^a.
To undo the exponentiation by b, we take the bth root (\sqrt[b]{\,\,}) of the equation above:
10^{a/b} = \sqrt[b]{10^a}.
1.7 the power 10
We used the base 10 in all the rules above, but there’s nothing sacred about it, we could have chosen any other number. Mastering how to work with powers of 10 will be extremely useful in the other chapters of this website, so that’s why you see it here.
1.8 power of a product / quotient
So far, we have kept the base fixed at 10 and explored how exponents combine. Now let’s look inside the base: what happens when we raise a product or a quotient to a power? The same idea, repeated multiplication, gives us two more rules.
Take, for instance
(ab)^3 =(ab)(ab)(ab) =(aaa)(bbb) =a^3b^3.
Each factor contributes one a and one b. If instead of 3 we had c factors, we get c copies of each:
(ab)^c = a^c b^c.
The quotient rule follows immediately by writing division as multiplication by a reciprocal:
\left(\frac ab\right)^c =(ab^{-1})^c =a^c(b^{-1})^c =a^cb^{-c} =\frac{a^c}{b^c},
… so in short,
\left(\frac ab\right)^c = \frac{a^c}{b^c}.
1.9 repeated multiplication
We started this chapter with repeated multiplication: for a positive integer n, 10^n means a product of n factors of 10. This gives us the product rule:
10^a \cdot 10^b = 10^{a+b}.
But what should 10^{-2} or 10^{3/2} mean? We can’t literally write a product containing -2 or 3/2 factors. Instead, we use the product rule as our guide. We define powers with zero, negative, and fractional exponents so that the rule keeps working. As we’ve seen, this leads us to 10^0=1, negative powers as reciprocals, and fractional powers as roots. Repeated multiplication gets us started, and preserving the rule takes us further.
1.10 to sum up
\begin{align} 10^a \cdot 10^b &= 10^{a+b} \\ 10^0 &= 1 \\ 10^{-a} &= \frac{1}{10^a} \\ 10^a \div 10^b &= 10^{a-b} \\ (10^a)^b &= 10^{ab} \\ 10^{a/b} &= \sqrt[b]{10^a} \\ (ab)^c &= a^c b^c \\ \left(\frac ab\right)^c &= \frac{a^c}{b^c} \end{align}