Powers raise a number or symbol to an exponent by multiplying it by itself. Negative powers produce fractions. The product, power, and quotient rules describe how to manipulate exponents when multiplying, raising to a power, or dividing terms with the same base. A root extracts a root of a number by raising it to a fractional exponent. Common and natural logarithms are two types that describe the power to which bases of 10 or e must be raised to obtain a given number. Logarithm rules allow logarithms of multiplied, divided, or raised terms to be expressed in terms of the logarithms of their components.

1. Power, Roots, Logarithms

2. Power
• In mathematics, that which is represented by
an exponent or index, denoted by a superior numeral. A
number or symbol raised to the power of 2 – that
is, multiplied by itself – is said to be squared (for
example, 32, x2), and when raised to the power of 3, it is
said to be cubed (for example, 23, y3). Any number to the
power zero always equals 1.
Powers can be negative. Negative powers produce
fractions, with the numerator as one, as a number is
divided by itself, rather than being multiplied by itself, so
for example 2-1 = 1/2 and 3-3 = 1/27.

3. Product Rule
The exponent "product rule" tells us that, when multiplying two powers
that have the same base, you can add the exponents. In this example,
you can see how it works. Adding the exponents is just a short cut!
Power Rule
The "power rule" tells us that to raise a power to a power, just multiply
the exponents. Here you see that 52 raised to the 3rd power is equal to
56.

4. Quotient Rule
The quotient rule tells us that we can divide two powers with the same
base by subtracting the exponents. You can see why this works if you
study the example shown.
Zero Rule
According to the "zero rule," any nonzero number raised to the power of
zero equals 1.
Negative Exponents
The last rule in this lesson tells us that any nonzero number raised to a
negative power equals its reciprocal raised to the opposite positive power.

5. Summary of index laws

6. Roots and Radicals
We use the radical sign :
It means "square root". The square root is actually a fractional index and is
equivalent to raising a number to the power 1/2.
So, for example:
251/2 = √25 = 5
You can also have
Cube root: (which is equivalent to raising to the power 1/3), and
Fourth root: (power 1/4) and so on.

7. Things to remember
If a ≥ 0 and b ≥ 0, we have:
However, this only works for multiplying. Please note that
does not equal
Also, this one is often found in mathematics:

8. Logarithms
Two kinds of logarithms are often used : common (or Briggian)
logarithms and natural (or Napierian) logarithms. The power to which a
base of 10 must be raised to obtain a number is called the common
logarithm (log) of the number. The power to which the base e (e =
2.718281828.......) must be raised to obtain a number is called
the natural logarithm (ln) of the number.
Log Rules:
1) logb(mn) = logb(m) + logb(n)
2) logb(m/n) = logb(m) – logb(n)
3) logb(mn) = n · logb(m)
4) logb = logb x1/y = (1/y )logb x