Chapter 5
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Transcript of Chapter 5
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Radical Expressions and Equations
CHAPTER 5
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Chapter 5:Radical Expressions & Equations
5.1 – WORKING WITH RADICALS
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RADICALS
radicand
index (is 2)
radical
Pairs of Like Radicals:
Pairs of Unike Radicals:
Mixed Radical: Entire Radical:
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EXAMPLE
Express each mixed mixed radical in entire radical form. Identify the values of the variable for which the radical represents a real number.
a) b) c)
a) b) c) Try it!
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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 which could be removed.• The radical is not part of the denominator of a
fraction.Not the simplest form, since 18 has 9 as a factor, whichis a perfect square.
Simplest form!
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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.
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EXAMPLE
Convert each entire radical to a mixed radical in simplest form.
a) b) c)
a) Find a factor that is a perfect square!
b) c) Try it!
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EXAMPLE
Simplify radicals and combine like terms.
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Independent Practice
P. 278-281 # 4, 6, 9, 10, 11, 13, 15, 17,
19, 21
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Chapter 5:Radical Expressions & Equations
5.2 – MULTIPLYING AND DIVIDING
RADICAL EXPRESSIONS
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MULTIPLYING RADICALS
When multiplying radicals, multiply the coefficients and multiply the radicands. You can only multiply radicals if they have the same index.
In general, , where k is a natural number, and m, n, a, and b are real numbers. If k is even, then a ≥ 0 and b ≥ 0.
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EXAMPLE
Multiply. Simplify the products where possible.
a)
b)
a) b) Try it!
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DIVIDING RADICALS
The rules for dividing radicals are the same as the rules for multiplying them.
In general, , where k is a natural number, and m, n, a, and b are real numbers. n ≠ 0 and b ≠ 0. If k is even, then a ≥ 0 and b > 0.
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RATIONALIZING DENOMINATORS
To simplify an expression that has a radical in the denominator, you need to rationalize the denominator. To rationalize means to convert to a rational number without changing the value of the expression.
Example: When there is a monomial square-root denominator, we can rationalize the expression by multiplying both the numerator and the denominator by the same radical.
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CONJUGATES
For a binomial denominator that contains a square root, we must multiply both the numerator and denominator by a conjugate of the denominator. Conjugates are two binomial factors whose product is the difference of two squares.• the binomials (a + b) and (a – b) are conjugates since their
product is a2 – b2
• Similarly,
So, they are conjugates!
• The conjugate of is
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EXAMPLES
Simplify each expression.
a) b)
a) Try it! b)
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CHALLENGE: TRY IT!
Simplify:
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Independent Practice
PG. 289-293, #4, 5, 10, 11, 13, 15, 17, 20, 21, 22, 23, 24,
26
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Chapter 5:Radical Expressions & Equations
5.3 – RADICAL EQUATIONS
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HANDOUT
You will need three metre sticks for the activity.
Answer all the questions to the best of your abilities, as this is a summative assessment. You will need to use your prior knowledge of radical expressions, as well as your understanding of triangles.
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SOLVING RADICAL EQUATIONS
When solving a radical equation, remember to:
• identify any restrictions on the variable• identify whether any roots are extraneous
by determining whether the values satisfy the original equation.
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EXAMPLE
a) State the restrictions on x in if the radical is a real number.b) Solve a) You can’t take the square
root of a negative number, so:
2x – 1 ≥ 0 2x ≥ 1 x ≥ ½
So, the equation has a real solution when x ≥ ½.
b)
x = 25 meets the restrictions, and we can substitute it into the equation to check that it’s a solution:
So, x = 25 is the solution.
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EXAMPLE
Solve
We need to check that they work:
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EXAMPLE
What is the speed, in metres per second, of a 0.4-kg football that has a 28.8 J of kinetic energy? Use the kinetic energy formula, Ek = ½mv2, where Ek represents the kinetic energy in joules; m represents mass, in kilograms; and v represents speed, in metres per second.
Ek = ½ mv2 Substitute in m = 0.4, and Ek = 28.8:
The speed of the football is 12 m/s.
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Independent Practice
P. 300-303 #7-10, 11, 13, 15, 17, 18, 19-23,
27