Mr. Suderman's Math Website
5.1: Working with Radicals
Consider the number 25: 25 = 52 and 25 = (–5)2.
So, 25 has two square roots: 5 and –5. 5 is called the principal square root of 25.
If we are taking a square root of a number, we must consider both the positive and negative square roots.
i.e. if: [pic]
[pic]
[pic]
Consider a square with area of 10.
The side length is the principal square root of 10; that is[pic]. Since 10 is not a perfect square, [pic] cannot be simplified and it is left as a radical.
[pic]
Convert Mixed Radicals to Entire Radicals
Example 1) Express each mixed radical in entire radical form. Identify the values of the variable for which the radical represents a real number.
[pic]
Your Turn
[pic]
Radicals in Simplest Form
A radical is in simplest form if the following are true:
• The radicand does not contain a fraction or any factor that can be removed.
• The radical is not part of the denominator of a fraction.
For example, [pic] is not in simplest form because 18 has a perfect square factor of 9, which can be removed. [pic] = [pic]= [pic] = [pic] .
Express Entire Radicals as Mixed Radicals
Example 2) Convert each entire radical to a mixed radical in simplest form.
[pic]
Your Turn
[pic]
Compare and Order Radicals
Example 3) Order the following numbers from least to greatest without a calculator.
Restrictions on Variables
If a radical represents a real number and has an even index, the radicand must be non-negative. The radical [pic] has an even index. So, 4 – x must be greater than or equal to zero.
Your Turn
Example 1) State the restrictions on the following radical expressions.
a) [pic] b) [pic]
Like Radicals
Radicals with the same radicand and index are called like radicals. When adding and subtracting radicals, only like radicals can be combined. You may need to convert radicals to a mixed form before identifying like radicals.
Add and Subtract Radicals
Example 4) Simplify radicals and combine like terms.
[pic]
Your Turn
Simplify radicals and combine like terms.
[pic]
Example 5) Consider the following designs shown for skateboard ramps. What is the exact distance across the base?
5.2: Multiplying and Dividing Radical Expressions
Objectives:
• Performing multiple operations on radical expressions
• Rationalizing the denominator
• Solving problems that involve radical expressions
Multiplying Radicals
When multiplying radicals, multiply the coefficients and multiply the radicands. You can only multiply radicals if they have the same index.
Ex. [pic] Radicals can be simplified before multiplying!
[pic]
Example 1) Multiply, and simplify the products where possible.
[pic]
Your Turn
Multiply. Simplify where possible.
[pic]
Dividing Radicals
When dividing radicals, divide the coefficients and then divide the radicands. You can only divide radicals that have the same index.
[pic]
[pic]
Rationalizing Denominators (We don’t want radicals in the final denominator!)
To simplify an expression that has a radical in the denominator, you need to rationalize the denominator, see the example below.
[pic]
For a binomial denominator that contains a square root, multiply both the numerator and denominator by a conjugate of the denominator.
Do you remember: (a – b)( a + b) = a2 – b2 , see the following example.
[pic]
[pic]
Example 2) Simplify each expression.
a) [pic] b) [pic]
Your Turn
Simplify each quotient.
a) [pic] b) [pic]
5.3: Radical Equations (Part I)
Radical Equations are equations with radical signs in them. As usual, we are trying to isolate the variable and determine possible values for the unknown.
Ex 1 a) State the restrictions on x in [pic] if the radical is to be a real number.
b) Solve [pic]
Your Turn
Identify any restrictions on y in [pic] if the radical is to be a real number. Then, solve the equation.
Example 2) Identify the restrictions on n in [pic] if the radical is to be a real number. Then, solve the equation.
Your Turn
Identify any restrictions on m in [pic]if the radical is a real number. Then, solve the equation. Check your solution(s).
To solve radical equations:
1. State any restrictions on the variables.
2. Isolate the radical & square both sides.
3. Solve the remaining quadratic equation.
4. Check your solution(s). Reject any extraneous roots.
*Recall: Extraneous roots are solutions that do not satisfy any initial conditions.
|5.3: Radical Equations (Part II) |
Example Solve [pic]. Check your solution.
Your Turn
Example Solve[pic]. Check your solution.
Objective: Modeling and solving problems with radical equations
Refer to the diagram of 3 metre sticks shown below. If the diagonal metre stick moves v cm down and h cm away from the wall, determine the dimensions of the right triangle. See textbook, p. 295.
a) If v = 10 cm, h = ____________.
b) If v = 40 cm, h = ____________.
Example What is the speed, in m/s, of a 0.4-kg football that has 28.8 J of kinetic energy? Use the kinetic energy formula, Ek = [pic]mv2, where Ek represents the kinetic energy, in joules; m represents mass, in kilograms; and v represents speed, in m/s.
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