1 Exponents and Radicals Chapter 11. 2 Flashback: One semester ago……..
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Transcript of 1 Exponents and Radicals Chapter 11. 2 Flashback: One semester ago……..
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Exponents and Radicals
Chapter 11
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RULES FOR INTEGRAL EXPONENTS: m, n positive real numbers
m nx x
0m
n
xx
x
( )m nx
( )nxy
0n
xy
y
0 0x x
0nx x
Flashback: One semester ago……..
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Simplifying Expressions with Integral ExponentsHelpful Tip: Watch the order of operations ….or else!
Examples
22
00
11
5 5
3 3
8 8
a a
m m
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Sect 11.1 : Simplifying Expressions with Integral ExponentsUse one or a combination of the laws of exponents to simplify each expression. State final answers using only positive exponents. Assume all variables represent positive values.
3 02 4 5 1 3
1) 3 2)2 4
x y x a b
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Simplifying Expressions with Integral Exponents
32 5 23) 4 4)m x y
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Simplifying Expressions with Integral Exponents
3
5
3 45) 6)
5b
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Simplifying Expressions with Integral Exponents
3 1 0
327)
2
y y y
y
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Simplifying Expressions with Integral Exponents
23
2 2
28)
3
ab
a b
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Simplifying Expressions with Integral Exponents
27
315
9) 10) 3
mz
n
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Simplifying Expressions with Integral Exponents
1 2211) 12) 3 2m n a b
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Simplifying Expressions with Integral Exponents
1 1
2 213)
x y
x y
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Simplifying Expressions with Integral Exponents
1 314) 3 2 x x y
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Sect 11.2: Fractional (Rational) Exponents
RATIO NAL EXPO NENTS:
nn xx 1
n mmnn
m
xxx
438
power
root
Note: These properties are valid as long as does not involve the even root of a negative number.
n a
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Fractional (Rational) Exponents
When evaluating expressions involving fractional exponents without a calculator:
It is usually best to find the root first, as indicated by the denominator, then raise it to the power indicated in the numerator.
2 33 427 16
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Fractional (Rational) Exponents
You can evaluate expressions involving fractional exponents using the calculator. Use the key and type parentheses around the fraction.
Example: Evaluate using the TI-84. 341296
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Simplifying Expressions Involving Fractional Exponents
The same laws of exponents apply to fractional exponents, though the work may be a little messier.
23
12
2 1
3 21)
2)
x x
x
x
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Simplifying Expressions Involving Fractional Exponents
4
2 33
12 23
12
3)
4) 64
x
x y
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Simplifying Expressions Involving Fractional Exponents
14
34
13
1
65)
2 6
6) 125
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Simplifying Expressions Involving Fractional Exponents
2 13 4
25
310
2
7) 1000 81
8)R R
R
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Sect 11.3: Simplifying Radicals
Let a and b represent positive real numbers.
nn n n
n n n
m n mn
n
nn
a a a
a b ab
a a
a a
bb
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Simplifying Radicals
Remember ….
is NOT equivalent to writingn n na b a b
9 16 9 16
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Simplifying Radicals
To reduce a radical to simplest form:
1. Remove all perfect nth power factors from a radical of order n.
2. If a fraction appears under the radical or there is a radical in the denominator of the expression, simplify by rationalizing the denominator.
3. If possible, reduce the order of the radical.
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Simplifying Radicals
Removing all nth power factors
3 18
3 5 12
1) 12
2) 54
x y
a b c
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Simplifying Radicals
Reducing the order of the radical
9
6 2 16
3) 27
4) 25a b
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Simplifying Radicals
Rationalizing the denominator
3
55)
2
26)
3
a
b
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Simplifying Radicals
Mixed bag…
6 7
2 1
7) 98
8)
x y
a b
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Sect 11.4: Addition and Subtraction of Radicals
To perform addition or subtraction of radicals, you combine like (similar) radicals.
Similar radicals differ only in their numerical coefficients (same radicand AND same index).
32 3 3 2 5 3 2
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Addition and Subtraction of Radicals
Plan of Action:
1. Express each radical in simplest form.
2. Combine like radicals.
1) 3 75 2 12 2 48 Examples
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Addition and Subtraction of Radicals
3 3 42) 32 50 18x x x
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Addition and Subtraction of Radicals
3 53 33) 24 3 4) 9 50 4 72x x x x x
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Addition and Subtraction of Radicals
25) 6 18
3
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Sect 11.5: Multiplication & Division of Radicals
To multiply expressions containing radicals, we will use the property
where a and b represent positive values.
Notice that the order (indexes) of the radicals being multiplied must be the same.
n n na b ab
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Multiplication of Radicals
Check it out!
1) 5 3 4 6
2) 4 2 3 5 2 8
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Multiplication of Radicals
3) 5 2 3 5 3
4) 7 3 2 7 3 2
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Division of Radicals
5
3 2Rationalize the denominator:
We saw earlier that when dealing with an expression containing a
radical in the denominator, we had to rationalize the denominator to
write it in simplest form.
We will now deal with denominators containing two terms.
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Division of Radicals
3 5
2 15 5
15Rationalize the denominator:
To rationalize a denominator that is the sum or difference of two terms, multiply the numerator and denominator of the fraction by the _____________________ of the denominator.
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Sect 14.4 Solving Radical Equations
2
33
nn
x x
x x
x x
To solve radical equations, we will use the fact that
That is, we will apply the appropriate inverse operation to “get the variable out of the radical”.
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Solving Radical Equations
To solve a radical equation involving one radical:
1. Isolate the radical expression on one side of the equation.
2. Raise both sides of the equation to the power that is the same as the order of the radical (inverse operation).
3. Solve the resulting equation for the variable.
4. Check for extraneous solutions* by checking the apparent solutions in the original equation.
*Extraneous solutions may be introduced when both sides of an equation are raised to an even power.
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Solving Radical Equations
2 2 3 10 16x
Example 1
Solve:
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Solving Radical Equations
4 1 8 9x x
Example 2
Solve:
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Solving Radical Equations
3- 7 7 -1 40 19n
Example 3
Solve:
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WIND POWER The power generated by a windmill is related to the velocity of the wind by the formula
where P is the power (in watts) and v is the velocity of the wind (in mph). Find how much power the windmill is generating when the wind is 29 mph.
Solving Radical Equations
3
0.02
Pv
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To solve a radical equation involving two square roots:
1. Isolate one of the radical expressions on one side of the equation.
2. Square both sides of the equation.
3. Simplify.
4. Isolate the remaining radical expression on one side of the equation.
5. Square both sides of the equation.
6. Solve the resulting equation for the variable.
7. Check for extraneous solutions by checking the apparent solutions in the original equation.
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Solving Radical Equations Involving Two Radicals
Example 1
Solve: 4d+1 6 1d
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Solving Radical Equations Involving Two Radicals
Example 2
Solve: 2 +5 2 1 4 0x x
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End of Section