Simplifying Radicals Notes - Rice University



Simplifying Radicals Notes

Objective: To simplify radicals.

I. Vocabulary

This is read the cube root of 6.

If the index number is not given then you are finding the square root. What would be the index number for a square root? ___________

II. To simplify

A. Perfect roots,

1. Find the root

2. Examples

a. [pic] =________

b. [pic] = ______

c. [pic] = ________

d. [pic]= ________

e. [pic]=_________

B. To finds the even (2,4,6…) roots of decimal numbers with even number of decimal places.

1. Divide the number of decimal places by the index number this tells you how many decimal places the root should have.

2. Take the root of the number

3. Examples

a. [pic] ________

b. [pic] ________

c. [pic]________

d. [pic] ________

C. Fractions that have the numerator and denominator as perfect roots.

1. Find the root of the numerator over the root of the denominator

2. Examples

a. [pic] = __________ c. [pic]= ___________

b. [pic] = __________ d. [pic]= ___________

D. Monomials that contain variables with exponents

1. Find the of the coefficient

2. Divide each exponent by the index number.

3. Examples

a.[pic]= _________ c. [pic]=__________

b. [pic]= _________ d. [pic]=_________

e. [pic] = _________ f. [pic] = _________

III. The "un-perfect" roots

A. Not all real numbers are perfect roots, so we must learn to simplify radicals.

B. Steps

1. Try looking for the hidden squares when taking the square root or hidden cube

root, etc. If you cannot spot it immediately then use the following:

2. Factor each number to its prime factors.

3. Look at the index use this number to group the like factors.

C. Examples

1. [pic]= [pic] since one of the twos has a twin one “two” can come out but one must stay behind. ___________

2. [pic] =[pic]= _________ this time we are looking for triplets

3. [pic]= ___________

D. Monomials with “un-perfect”

1. Steps

a. Simplify the coefficient.

b. Divide each exponent by the index number.

c. The quotient comes out with the variable and the remainder must stays under the behind under the radical sign.

2. Examples

a. [pic] = _____________

b. [pic] = _____________

c. [pic] = _____________

d. [pic]= ____________

E. Rational numbers (fractions with perfect root denominators)

1. Steps

a. Reduce the fraction if possible.

b. Take the root of the denominator.

c. Simplify the radical in the numerator.

2. Examples

a. [pic] = _________

b. [pic]= _________

c. [pic] = _________

Adding or Subtracting Radical Expressions

Objective: To add or subtract radical expressions.

I. Radical addition and subtraction

A. Steps and rules

1. RULE: The radicands and the index numbers must be equal before you can add or subtract the numbers on the outside.

2. Simplify each radical first; otherwise you will not know if the radicands are equal.

3. Only add or subtract the numbers on the outside DO NOT change the radicands.

B. Examples

1. [pic] What changed? What stayed the same?

2. [pic] ___________________

3. [pic] _______________

Multiplying Radicals

Objective: to multiply two or more radicals and simplify answers.

I. Radical multiplication

A. Rules and steps for monomials

1. Index numbers must be the same

2. Multiply radicands to radicands (they do not have to be the same).

3. Multiply outside numbers to outside numbers

4. SIMPLIFY, SIMPLIFY, SIMPLIFY!

B. Rules and steps for a monomial times a binomial

Distribute and follow the rules for monomial radicals

C. Rules and steps for binomials

Use FOIL and follow the rules for monomial radicals

D. Examples

1. [pic] _______________

2. [pic] _______________

3. [pic] _______________

4. [pic] _______________

5. [pic] _______________

6. [pic] _______________

7. [pic] _______________

What did you notice about the last example? This is a very important concept that will be used in the next lesson. Notice the terms.

It is the same as the difference of two squares.

Square the first term minus the square of the last term.

[pic]

Division of Radicals and Rationalizing the Denominator Notes

Objective: to simplify the quotients of radical expressions and rationalize the denominators.

Do you know what these equal? [pic] = ________ [pic]=__________

What is a rational number?

I. To simplify quotients of monomial radical expressions

A. Rule

Do not leave a radical in the denominator EVER!!!!!!!

And

Index numbers must be the same.

B. Steps

1. Write the expression under one radical sign

2. Simply the fraction, if possible

3. Re-separate into a radical on top and radical on the bottom

4. Simplify each radical

5. Rationalize the denominator.

Look at the denominator only!

Multiply top and bottom by the radical that makes the denominator rational.

C. Examples

1. [pic] 2. [pic]

3. [pic] 4. [pic]

5. [pic] 6. [pic]

II. Simplifying radicals that have binomials in the denominators.

A. Conjugates are binomial radical expressions that have the same terms but different signs.

NOT JUST THE RADICLAL IN THE DENOMINATOR!!!

Example: [pic]=

B. Steps

1. Simplify each radical if possible

2. Multiply the numerator and denominator by the conjugate of the denominator.

3. Simplify the radicals again.

4. Factor any common term IF ANY!

5. Reduce the factor of the numerator and denominator if possible.

C. Example

1. [pic] 2. [pic] 3. [pic]

conjugate __________ conjugate __________ conjugate __________

Radical Equations Notes

Objective: to solve radical equations

I. Solving radical equations

A. Steps

1. Index numbers must be the same. If there are radicals on both sides of the equal signs.

2. ISOLATE THE RADICAL!!! If possible.

3. Combine like terms if necessary

4. Raise both sides to the power indicated by the index number to get rid of the radical sign

5. Solve

6. Check, you might have extraneous solutions. You will miss the problem if you leave an extraneous solution. So check it!!

B. Examples

1. [pic] Steps 1.

2.

3.

4. Check even if you hate to do it because you can get extraneous solutions.

2.[pic] 3. [pic] 4. [pic]

5. [pic] 6. [pic]

Pythagorean Theorem Notes

In a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. 1. 2.

The Distance Formula Notes

Objective: to find the distance between two given points

I. The distance formula is derived from the Pythagorean Theorem.

Formula: [pic] memorize!

II. Find the distance between the given points.

S(-3,2);T(5,1)

[pic]

Midpoint Notes

Objective: To find the midpoint of a line segment.

I. The formula for midpoint

A. [pic] = the x coordinate for the midpoint

[pic] = the y coordinate for the midpoint

Memorize the midpoint formula, M [pic]

In other words find the average of the x-coordinates for the two end points and the average of the

y-coordinates for the endpoints.

B. Examples

1. Find the midpoint given the following endpoints:

A(-5,5) and B (4,-6).

[pic] so the midpoint is [pic]

2. Find the endpoint if the midpoint is (3,4) and one endpoint is (-5,7)

[pic] [pic] the other endpoint is (11,1)

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

B

[pic]

c

Memorize!!! This will really save you time.

C

A

b

a

[pic]

x 5

4

x 2

6

[pic]

[pic]

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