## Archive for June, 2014

## Definitions and Rules for Exponents

**Definitions and Rules for Exponents**

For all integers *m *and *n *and all real numbers *a *and *b*, the following rules apply.

Product Rule a^{m}.a^{n} = a ^{m+n}

Quotient Rule = a^{m-n}(a ≠ 0)

Zero Exponent a^{0} = 1 (a ≠ 0)

Negative Exponent a ^{–n } = (a ≠ 0)

Power Rules (a^{m})^{n }= a^{mn}

(ab)^{m } = a^{m}b^{m}

Special Rules

## Polynomial Function

**Polynomial Function**

A **polynomial function of degree **** n **is defined by

f(x) = a_{n}x^{n }+ a_{n-1} x^{n-1} + ….. +a_{1}x + a_{0,..}

for real numbers a_{n}, a_{n-1},…..a_{1}, and a_{0} , where a_{n} ≠ 0 and n is a whole number.

## Polynomial

**Polynomial**

A **polynomial **is a term or a finite sum of terms in which all variables

have whole number exponents and no variables appear in denominators

or under radicals.

Examples of polynomials include

3x – 5 , 4m^{3} – ^{5m2p} + 8

## Converting to Scientific Notation

**Converting to Scientific Notation**

** **

** Step 1 :- **Position the decimal point. Place a caret, ^, to the right of the

first nonzero digit, where the decimal point will be placed.

* Step 2:- *Determine the numeral for the exponent. Count the number

of digits from the decimal point to the caret. This number gives

the absolute value of the exponent on 10.

** Step 3:- **Determine the sign for the exponent. Decide whether multiplying

by 10^{n}should make the result of Step 1 larger or smaller. The

exponent should be positive to make the result larger; it should

be negative to make the result smaller.

**Converting from Scientific Notation**

Multiplying a number by a positive power of 10 makes the number larger,

so move the decimal point to the right *n *places if *n *is positive in 10* ^{n}*.

Multiplying by a negative power of 10 makes a number smaller, so

move the decimal point to the left places if │*n│ *is negative.

If *n *is 0, leave the decimal point where it is.

## Power Rules for Exponents

**Power Rules for Exponents**

If *a *and *b *are real numbers and *m *and *n *are integers, then

**(a) **(a^{m})^{n }= a^{mn} .

**(b) **(ab)^{m} = a^{m}b^{m}

**(c) **(a/b)^{m} = a^{m}/b^{m} (b ≠0).

In words,

**(a) **to raise a power to a power, multiply exponents;

**(b) **to raise a product to a power, raise each factor to that power; and

**(c) **to raise a quotient to a power, raise the numerator and the denominator

to that power.

## Quotient Rule for Exponents

**Quotient Rule for Exponents**

If *a *is any nonzero real number and *m *and *n *are integers, then

a^{m}/a^{n} = a^{m-n.}

In words, when dividing powers of like bases, keep the same base and

subtract the exponent of the denominator from the exponent of the

denominator from the exponent of the numerator.

## Define 0 and negative exponents

**Define 0 and negative exponents**

** **

** **So far we have discussed only positive exponents. Now we define 0 as an exponent. Suppose

we multiply 4^{2}by 4^{0}. By the product rule, extended to whole numbers,

4^{2} . 4^{0} = 4^{2+0} = 4^{2}.

For the product rule to hold true, 4^{0}must equal 1, and so we define *a*^{0}this

way for any nonzero real number *a*.

**Zero Exponent**

If *a *is any nonzero real number, then

*a*** ^{0}**=

**1.**

*The expression 0*^{0}** is undefined**.*

**Negative Exponent**

For any natural number *n *and any nonzero real number *a*,

a^{-n} = 1/a^{n}.

## Product Rule for Exponents

**Product Rule for Exponents**

** **

If *m *and *n *are natural numbers and *a *is any real number, then

* a ^{m}. a^{n} = a^{m+n}.*

** **

In words, when multiplying powers of like bases, keep the same base

and add the exponents.

## Use row operations to solve a system with two Equations

**Use row operations to solve a system with two Equations **

** **

Row operations can be used to rewrite a matrix. The goal is a matrix in the form

.

for systems with equations, respectively. Notice that there are 1s down the diagonal from upper left to lower right and 0s below the 1s. A matrix written this way is said to be in **row echelon form. **When these matrices are rewritten as systems of equations, the value of one variable is known, and the rest can be found by substitution. The following examples illustrate this method.

## Matrix Row Operations

**Matrix Row Operations**

** **

**a.** Any two rows of the matrix may be interchanged.

**b. **The numbers in any row may be multiplied by any nonzero real

**c. **Any row may be transformed by adding to the numbers of the row the product of a real number and the corresponding numbers of another row.

Examples of these row operations follow.

Row operation 1:

becomes . Interchange row 1 and row 3.

Row operation 2:

. Multiply the number in row 1 by 3.

Row operation 3:

. Multiply the number in row 3 by -2; add then to the corresponding numbers in row 1.

The third row operation corresponds to the way we eliminated a variable from a pair of equations in the previous sections.