Chapter 4Radicals

4.1 – Mixed and Entire Radicals

Chapter 4: Radicals

Review:Squares and Square Roots

Remember: Squares and Square Roots are opposite operations.

The principal square root is the positive square root of a number.

The secondary square root is the negative square root of a number.

Unless specified, typically a square root symbol is only looking for the principal square root.

Example: Simplifying

a)

b)c)

When you’re simplifying radicals, the first step is to factor the number into its prime factors.

Example: Expanding

a) b) c)

When turning a mixed radical into an entire radical, you must first pull the outside coefficient in underneath the radical sign. Remember, roots and exponents are opposite operations!

pg. 182-183, #1, 2, 3, 5, 6, 7, 11, 12, 13, 16, 18, 20Independent practice

4.2 – Adding and Subtracting

Radicals Chapter 4: Radicals

Example

Determine the length of fascia needed.

Karen’s uncle needs 9 metres of fascia.

Check on your calculator! What is the answer different?

ExampleDetermine the difference in length between each pair of sides.a) PS and SRb) RQ and PQa) Let D represent the difference between PS and SR

The difference between PS and SR is cm.

Why is our answer positive?

b) Let E represent the difference between RQ and PQ

The difference between RQ and PQ is cm.

Example

a) b)

Your turn…

Create a negative mixed radical using one addition sign, one subtraction sign, and the radicals , and . Challenge the person sitting next to you to simplify it!

The Rules

• You can take a number out from under a root sign by square-rooting (or cube-rooting, if it’s a cubed root) the number. • Ex:

• You can bring a number inside the a root sign by squaring (or cubing, if it’s a cubed root) the number.• Ex:

• You can add or subtract like radicals.• Ex:

• You cannot add or subtract unlike radicals.

pg. 188-190, #1, 2, 3, 5, 6, 8, 9, 11, 13, 15, 16, 18, 19 Independent practice

4.3 – Multiplying and

Dividing Radicals

Chapter 4: Radicals

Rules for multiplication and division

The product of two square roots is equal to the square root of the product.

i.e.

The quotient of two square roots is equal to the square root of the quotient.

i.e.

Prove:

• What is a square root?• Is there a way to represent

a square root with exponents?

Recall that exponent laws dictate that ax bx=(ab)x

Now, try to prove the division rule on your own:

ExampleSimplify:

Try this!

Express the following expression in its simplest form:

Example

Simplify:

Method 1:

Method 2: (Rationalizing the denominator)

Method 2 is called rationalizing the denominator, which is done by multiplying both the numerator and the denominator by the radical in the denominator.

Example continued…

Method 3: (Finding a common factor)

Simplify:

• Which method seemed easiest to you?

• What might be some advantages to each method?

• What might be some disadvantages?

• Will all the methods work all of the time?

pg. 198-200, #1-5, 8-16, 19-21Independent practice

Handout

Today we will be working on an project about Pythagorean fractal trees. Make sure to answer all the questions to your fullest ability, as this is a summative assessment. You may need a ruler for parts of the assignment.

4.4 – Simplifying Algebraic

Expressions Involving Radicals

Chapter 4: Radicals

If an algebraic expression involves a radical like , can x be any real number?

We sometimes have to apply restrictions to variables. Restrictions are the values of the variable in an expression that ensure the expression in defined.

Ex. For 1/x, x cannot be equal to zero, or symbolically we can write x ≠ 0.

What about:• • •

Example: Identify the restrictions

a) is definedwhenwhere

b) is definedwhen

c) is definedwhen where

Remember you can’t take the square root of a negative number

Square roots and squares are opposites, so they cancel to leave the absolute value of x.

Simplify the expression, and remember that you can’t take the square root of a negative.

Can we make any general rules about the restrictions on radicals in algebraic expressions? Consider .

Example

a) is definedwhenwhere

You must always identify your restrictions when simplifying algebraic expressions involving radicals.

b) is definedwhen

Example

Simplify:

is defined when y ≥ 0, where y R.

Example

Simplify:

is defined when x ≥ 0, where x R.

Remember, while multiplying radical expressions, you multiply the coefficients by the coefficients, and combine the radicals.

ExampleSimplify:

is defined for x > 0, where x R.

What about an expression like ? What are the restrictions?

Just remember that the number underneath the radical needs to stay positive.

pg. 211-213, #1, 3, 6, 8, 9, 10, 11, 13, 14, 15, 17Independent practice

4.5 – Exploring Radical

EquationsChapter 4: Radicals

Solving Radical Equations

What is the opposite of a square root?

Ex.

To get rid of a cube root, you need to cube both sides.

Make sure that you simplify the equations as much as possible before solving. Remember your orders of operations.

Try it!

Solve for x:

pg. 215, #1-5 Independent practice

4.6 – Solving Radical

EquationsChapter 4: Radicals

Solving Radical Equations

If the solution does not work when substituted back into the equation, then it is an extraneous root, and the equation does not have a solution.

Ex. is defined for x ≥ -2, where x R. because

Check by substitution:Since 3 ≠ -3, this is an extraneous root.

You always must check your solution when solving radical equations.

ExampleThe forward and backward motion of a swing can be modeled using the formula

where T represents the time in seconds for a swing to return to its original position, and L represents the length of the chain supporting the swing, in metres. When Cara was swinging, it took 2.5 s for the swing to return to its original position. Determine the length of the chain supporting her swing to the nearest centimetre.

Substitute:

Solution:

The length of the chain is 1.55 metres, or 155 cm.

Alternate Solution

Another way of solving these types of problems is to first substitute in your values, and then manipulate the equation.

Solution #2:

The solution is the same!

• Which method do you prefer?• What are some of the advantages

and disadvantages of each?• What if you had to solve multiple

problems using the same equation?

pg. 222-224, #1-15Independent practice