*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 *x*^{2 }= *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?