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7.1 and 7.2 Simplifying/Multiplying/Dividing Radical expressions. Nth Roots. For any real number a and b, and any positive integer n, if a n = b , then a is the n th root of b. Real Number Examples :. Find the roots: Square Root of 4 Square Root of - 4 - PowerPoint PPT Presentation
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7.1 and 7.2 Simplifying/Multiplying/Dividing Radical expressions
+Nth Roots
+
For any real number a and b, and any positive integer n, if an = b , then a is the nth root of b.
+Real Number Examples:
Find the roots:
1. Square Root of 4 Square Root of -4
2. Cube Root of 8 Cube Root of -8
3. Fourth Root of 16 Fourth Root of -16
4. Fifth Root of 32 Fifth Root of -32
+Variable Examples:
Find the Square Root of:
1. a1 2. a2
3. a3 4. a4
+
+
When a number has two real roots, the positive root is called the principal root.
We always use the principal root when simplifying.
+Examples – Simplify :
1. 2.
+
3. 4.
+
5. 6.
+7.
8.
9.
+10.
11.
12.
+
Multiplying and Dividing Radical Expressions
If they are real numbers, then
+Examples:
1.
2.
+
3.
4.
+
5.
6.
+Dividing Radical Expressions
+Examples:
1. 2.
+Rationalizing the Denominator
**Multiply the numerator and denominator by the denominator**
Then Simplify
Example: 1.
+
2.
3.
+
4.
5.
+
7.3Adding, Subtracting, Multiplying and Dividing Binomial Radical Expressions
+Adding Radical Expressions
Use the same concept as that of adding or subtracting like variables.
Example: 7 - 3x + 2x + 5
*Have to have like Terms to Add/Subtract*
+Like Radicals are radical expressions that have the same index and the same radicand.
xx 23
+
Like Radicals Unlike Radicals
= =
+Examples:
1.
2.
2624
425257
+3.
4.
5.
6.
55363
7875
2726
28 xx
+Always simplify radicals before combining!
1. 2.
+3.
4.
5.
6.
+Multiplying Radical Expressions
When multiplying radicals, one must multiply the numbers OUTSIDE (O) the radicals AND then multiply the numbers INSIDE (I) the radicals.
83182
+Dividing Radical Expressions
When dividing radicals, one must divide the numbers OUTSIDE (O) the radicals AND then divide the numbers INSIDE (I) the radicals.
Remember to rationalize the denominator if needed!
+Examples:
1. 2.
432412
58248
+Multiplying Binomials
To multiply, USE FOIL!
Example 1:
+
2.3.
)31)(31(
+Dividing Binomial Radicals
To divide, Rationalize the denominator!
(a + b)( a - b) = a2 – b2
These are called conjugates! They make radicals disappear!
+Examples:
1.
+
2.
+Solve:
1.
2.
+
3.
4.
+Examples:
1.
+
2.
+
3.
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4.
+Homework
Practice: Practice 7-3 #1-30 Left Column
+
7.4 Rational Exponents
+Homework Check
+Rational Exponents
+
Rational Exponents are another way to write radicals.
+Simplify each expression.
1.
2.
+
3.
+
4. 5.
+
6.
+
+Converting to Radical Form
1.
2.
+
3. 4. 5.
+Converting to Exponential Form
1.
2.
+
3. 4. 5.
+
Properties of Exponents also apply to Rational Exponents!
Write in Radical Form:
+2.
3.
4.
5.
+Simplify each expression.
1.
2.
+
3.
4.
+Homework
Practice 7-4 # 1-25, 29-44 Left Column ONLY
+
7.5 Solving Radical Equations
+GBMP Review
1. y=0.5(2)^x 2.130% 3. 0.69
4. 32^(3/5)=8 5. x=43.3
6. ln(a^3/b^5) 7.x=0.1879 8. -2
9. X = 0.2693 10. $7708.03
11. $119.94 12. y = 250(1.60)^x
+Homework Check
+Radical Equations
A radical equation is an equation that has a variable in a radicand or has a variable with a rational exponent.
Are these Radical Equations?
+We use inverse operations to solve equations.
Solve: X2 = 4
The inverse of squaring a function is finding the square root.
+
Solve: X3 = 64
The inverse of cubing a function is finding the cube root.
+
The inverse of raising a function to the nth power is finding the nth root.
+Solve the following. Check your solutions!
1. 2.
+
3. 4.
+5.
+6.
+Solve (x)1/2 = 3
Recall that ½ = , so we square both sides!
What do you know about 2 and ½?
**To solve radical equations with rational exponents, raise each side to the reciprocal
exponent!
+
n
m
kx
………
…
Therefore,
+Examples:
1. 2.
+
3. 4.
+
Remember if the numerator is squared then we must do ±
1. 2.
+
3. 4.
+
You may also solve radical equations by graphing! Set the equation equal to zero and graph.
The solutions are the x-intercepts!
+
Given:
The equation is already equal to zero!
y=
Use the zero function to find the x-intercepts!