header
header2



ACCELERATED PRE-CALCULUS SUMMER STUDY GUIDE

CHAPTER 9: ROOTS


SECTION 9.1: RADICALS, INDEXES, RADICANDS

The following symbol, $\sqrt{}$ , is called a "radical" symbol or a "root" symbol. In general, you'll see a number inside this little "v" portion of the radical (root) symbol. That number inside the v portion of the radical (root) symbol is called an "index." The math expression "inside" the radical symbol is called the "radicand."

Figure

A radical with an index of 2, $\root{2} \of{}$ is called a "square root." In general, math people do not place a 2 in the index position: if you see $\sqrt{}$ it is understood to be a "square root." A radical with an index of 3, $\root{3} \of{}$ is called a "cube root." A radical with an index of 4, $\root{4} \of{}$ is called a "fourth root." A radical with an index of n, $\root{n} \of{}$ is called a "nth root."

EXAMPLES OF ROOTS
root2 pronounced "square root of 2"
root3 pronounced "cube root of 5"
root4 pronounced "fourth root of 9"
root5 pronounced "fifth root of 7"
root17 pronounced "seventeenth root of 9234"

 

SECTION 9.2: TWO INDEX RULES OF RADICALS

 

ROOTS WITH INDEXES THAT ARE POSITIVE EVEN INTEGERS

If "a" is a positive even integer and "b" is a positive real number

then $\root{a} \of{b}$ means

"What positive real number raised to the power of "a" will result in "b"?"

 

ex. 1) √ 4 is asking "What positive real number raised to the power of 2 will result in 4?"

The answer is 2.
Why? Because 22 = 4. Hence, we say √ 4 = 2.

Comment: Rememeber if there is no index in the root, the root is assumed to have an index of 2.

ex. 2) cube8 is asking "What positive real number raised to the power of 4 will result in 81?"

The answer is 3.
Why? Because 34 = 81. Hence, we say cube8= 3.

ex. 3) cube8 is asking "What positive real number raised to the power of 6 will result in 64?"

The answer is 2.
Why? Because 26 = 64. Hence, we say cube8= 2.

 

ROOTS WITH INDEXES THAT ARE POSITIVE ODD INTEGERS

If "a" is a positive odd integer and "b" is any real number

then $\root{a} \of{b}$ means

"What real number raised to the power of "a" will result in "b"?"

Comment: In this case your answer will be positive or
negative depending on the radicand.


ex. 1) cube8 is asking "What real number raised to the power of 3 will result in 8?"

The answer is 2.
Why? Because 23 = 8. Hence, we say cube8= 2.

ex. 2) cube8 is asking "What real number raised to the power of 5 will result in -32?"

The answer is -2.
Why? Because (-2)5 = -32. Hence, we say cube8= -2.

Comment: Roots with odd indexes can have radicands that are negative and answers that are negative.

 

 

SECTION 9.3: ROOTS AS EXPONENTS

There is rule that allows you to transform a radical into an exponential. It is as follows:

RADICAL TO EXPONENT RULE

MATH


ex. 1) cube8 can be rewritten as 8 1/3.

ex. 2) cube8 can be rewritten as (-32) 1/5.

ex. 3) 481 can be rewritten as (81) 1/4.

ex. 4) 481 can be rewritten as (32) 1/6.

 

Figure

 

 

-----------------------------
THE SUMMER STUDY GUIDE
BY CHAPTERS

-----------------------------

RETURN TO THE SUMMER STUDY GUIDE MAIN PAGE

- CHAPTER 1: THE NUMBER SYSTEM

- CHAPTER 2: ORDER OF OPERATIONS

- CHAPTER 3: VARIABLES, MONOMIALS, BINOMIALS, TRINOMIALS, POLYNOMIALS,
COEFFICIENTS, TERMS AND LIKE TERMS

- CHAPTER 4: SIGNED NUMBERS, ABSOLUTE VALUE, AND INEQUALITY SYMBOLS

- CHAPTER 5: FACTORS, COMMON FACTORS, LEAST COMMON FACTORS AND GREATEST COMMON FACTORS

- CHAPTER 6: PROPERTIES OF NUMBERS

- CHAPTER 7: THE WORLD OF FRACTIONS

- CHAPTER 8: EXPONENTS

- CHAPTER 9: ROOTS

- CHAPTER 10: ALGEBRAIC EXPRESSIONS

- CHAPTER 11: CARTESIAN COORDINATE SYSTEM

- CHAPTER 12: SETS, RELATIONS AND FUNCTIONS

- CHAPTER 13: AVERAGE RATE OF CHANGE OF Y WITH RESPECT TO X, SLOPE, PYTHAGOREAN THEOREM, AND DISTANCE FORMULA BETWEEN TWO POINTS

- CHAPTER 14: X-INTERCEPT(ZERO) AND Y INTERCEPT(B)

- CHAPTER 15: LINES

- CHAPTER 16: FUNCTIONS

- CHAPTER 17: MULTIPLYING POLYNOMIALS

- CHAPTER 18: FACTORING

- CHAPTER 19: RATIONAL EXPRESSIONS

- CHAPTER 20: SOLVING EQUATIONS

- CHAPTER 21:SOLVING INEQUALITIES

- CHAPTER 22: SOLVING A SYSTEM OF EQUATIONS

- CHAPTER 23: QUADRATICS

- CHAPTER 24: CIRCLES

- CHAPTER 25: AREAS AND PERIMETERS OF PLANE FIGURES

- CHAPTER 26: VOLUMES