Operations with powers.
1. At multiplying of powers with the same base their exponents are added:
a m ·a n = a m + n
2. At dividing of powers with the same base their exponents are subtracted:
3. A power of product of two or some factors is equal to a product of powers of these factors:
( a·b·c… ) n = a n · b n · c n …
4. A power of a quotient (fraction) is equal to a quotient of powers of a dividend (numerator) and a divisor (denominator):
(a / b ) n = a n / b n
5. At raising of a power to a power their exponents are multiplied:
(a m ) n = a m n
6 . A power of some number with a negative (integer) exponent is defined as unit divided by the power of the same number with the exponent equal to an absolute value of the negative exponent:
Example: ( 2 · 3 · 5 / 15 ) 2 = 2 2 · 3 2 · 5 2 / 15 2 = 900 / 225 = 4
Operations with roots.
In all below mentioned formulas a symbol means an arithmetical root ( all radicands are considered here only positive ).
1. A root of product of some factors is equal to a product of roots of these factors:
2. A root of a quotient is equal to a quotient of roots of a dividend and a divisor:
3. At raising a root to a power it is sufficient to raise a radicand to this power:
4. If to increase a degree of a root by n times and to raise simultaneously its radicand to the n-th power, the root value doesn’t change:
5. If to decrease a degree of a root by n times and to extract simultaneously the n-th degree root of the radicand, the root value doesn’t change:
Square Roots and Other Radicals
In this section, we will review some facts about square roots and other radicals which are relevant in calculus.
Here is the formal definition of a square root.
means that x2 = a, for x >= 0.
That is, x is the non-negative number whose square is a. For example,
since (0.3)2 = 0.09.
So to find the domain of a function with a quadratic expression under the root sign (that is, the radicand is quadratic), one might have to solve a quadratic inequality.
Properties of square roots
Square roots have the following properties.
,where a, b >= 0
,where a >= 0 and b > 0
,for any real number a
, (note that a >= 0 here)
It is important to notice that, unlike the product property for square roots
,
there is no similar rule for a sum under a square root. That is, in general
.
Square roots in denominators
When you are adding or terms which contain square roots in a denominator, you may find it helpful to write the expression as a single fraction. This involves using techniques from algebra, such as finding a common denominator, which is shown in the following example. (Here 
, is the common denominator.)
Similar techniques can be used when the expression under the root sign has more than one term.
Rationalizing
“Rationalizing” a denominator involves eliminating (algebraically) a square root from the denominator of an expression. When the denominator has two terms, we rationalize by multiplying numerator and denominator by the conjugate of the denominator.
In this example, can you identify the conjugate?
