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 am.an = a m+n

Quotient Rule \frac{a^{}m}{a^{n}} = am-n(a ≠ 0)

Zero Exponent a0 = 1 (a ≠ 0)

Negative Exponent a –n = \frac{1}{a^{n}} (a ≠ 0)

 

Power Rules (am)n = amn

(ab)m = ambm

(\frac{a}{b})^{m}= \frac{a^{m}}{b^{m}} (b \neq 0)

 

Special Rules \frac{1}{a^{-n}} = a^{n}(a \neq 0) \frac{a^{-n}}{b^{-m}} = \frac{b^{m}}{a^{n}} (a,b\neq 0)

a^{-n} = (\frac{1}{a})^{n}(a\neq 0)(\frac{a}{b})^{-n}=(\frac{b}{a})^{n}(a,b\neq 0)

 

Polynomial Function

Polynomial Function

A polynomial function of degree n is defined by

f(x) = anxn + an-1 xn-1 + ….. +a1x + a0,..

for real numbers an, an-1,…..a1, and a0 , where an ≠ 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 , 4m35m2p + 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 10nshould 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 10n.

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)  (am)n = amn .

(b)  (ab)m = ambm

(c)  (a/b)m = am/bm (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

am/an = am-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 42by 40. By the product rule, extended to whole numbers,

42 . 40 = 42+0 = 42.

For the product rule to hold true, 40must equal 1, and so we define a0this

way for any nonzero real number a.

Zero Exponent

If a is any nonzero real number, then

a0= 1.

The expression 00is undefined.*

 

Negative Exponent

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

a-n = 1/an.

 

Product Rule for Exponents

Product Rule for Exponents

 

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

 

   am. an = am+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

\begin{bmatrix}  1 & a& \left.\begin{matrix}  &  & \\  &  &  \end{matrix}\right|b\\  0& 1 & \left.\begin{matrix}  &  & \\  &  &  \end{matrix}\right|c  \end{bmatrix}.

 

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:

\begin{bmatrix}  2 &3  &9 \\  4&8  &-3 \\  1& 0 & 7  \end{bmatrix} becomes \begin{bmatrix}  1 & 0 &7 \\  4& 8 & -3\\  2& 3 & 9  \end{bmatrix} . Interchange row 1 and row 3.

 

Row operation 2:

 

\begin{bmatrix}  2 &  3&9 \\  4 & 8 &-3 \\  1& 0 & 7  \end{bmatrix} becomes \begin{bmatrix}  6 &  9&27 \\  4& 8 &-3 \\  1&0  & 7  \end{bmatrix}. Multiply the number in row 1 by 3.

 

Row operation 3:

 

\begin{bmatrix}  2 & 3 &9 \\  4 & 8 & -3\\  1& 0 & 7  \end{bmatrix} becomes \begin{bmatrix}  0 & 3 &-5 \\  4& 8 & -3\\  1& 0 & 7  \end{bmatrix}. 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.