Basics

Real NumBeRs

of

coNt.

Algebra

Exponents and nth Roots

Exponents

nth Roots

Exponent is a short-hand notation for repeated multi-

The nth root is the inverse operation of raising a number

plication.

to the nth power. So the inverse operation of xn = y is

.

•

·

2 · 2

2 = 23. We say that 2 is raised to the power

•

of 3.

= 4 because 42 = 16

•

is called the radical sign

•

·

2 · 2 · 2 · 2

2 = 25. We say that 2 is raised to the

•

When n=2, we usually write

, not

, and we call

power of 5.

it the square root.

•

For x , we say that x is raised to the power of n.

n

•

When n=3, we call it the cube root.

• x

and n are variables, symbols that are used to

represent a value.

If the nth root can’t be simplified (reduced) to a rational

•

If n=2, we can also say x squared. If x=3, we

number without the radical sign (

), the number is

irrational.

say x cubed.

•

Examples:

= 8, so it is a rational number.

cannot be reduced any further and is irrational.

We can get an approximate value for irrational square

roots by using the calculator. In decimal form, the

number will look like an unending string of numbers.

•

Example:

≈ 1.414 when rounded to three decimal

places.

Fractions and Decimals

A rational number is just a ratio of one number to another written in fraction form as

.

•

Every whole number is a rational number where the denominator equals 1.

•

A denominator equal to 1 is sometimes called the invisible denominator because it is not usually written out:

.

•

Fraction bar: the line that separates the numerator and the denominator. The denominator b ≠ 0.

•

A proper fraction represents a number less than one because a < b, while an improper fraction represents a

number greater than one because a > b.

•

A negative fraction is usually written with the negative sign to the left of the fraction

•

Example:

could equal

or

•

Improper fractions can be rewritten as an integer plus a proper fraction (mixed number).

•

Example:

•

Whenever we can write two fractions equal (=) to each other, we have equivalent fractions.

•

Example:

, so

and

are equivalent fractions.

•

The fractions

are equivalent as long as c ≠ 0.

A fraction can be converted into a decimal - just divide the numerator

by the denominator.

Figure: Equivalent fractions

•

Examples:

•

The ... means that the decimal goes on forever.

Not all decimals can be converted into fractions.

•

If the numbers after the decimal point (.) never repeats and never ends, the number is irrational.

•

Any number that can’t be written as a fraction is irrational.

Page 2 of 3