Download MAT 1033 - Chapter 10 - Operations on Radicals

MAT 1033
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Multiplying and Dividing Radicals (square roots)
To multiply radicals
1. Multiply the coefficient and multiply the radicands (the expression inside the radical)
2. Simplify your answers
Note: You may simplify before you multiply.
Examples:
1. 3  2  
2.
3 
3 
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3.
5  15  
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. 2 7 x  3 14 x
Exercises: Multiply (Distribute). Simplify your answers
1.
3(1 +
2)
2.
2 (2 - 3 2 )
3.
3 (4 +
6)
4.
5 ( 2 3 - 3 10 )
5. (4 +
6. ( 3  2 ) ( 3  2 )
3 ) (5 - 2 3 )
8. (2 5  3)
7. (5  2 7 ) ( 5  2 7 )
2
To divide radicals:
a
Use the property
=
b
a
b
and the following steps to simplify radical expressions.
Steps:
1. Reduce fractions within the radical if possible.
2. Remove any “perfect roots” from under the radical.
3. If a radical remains in the denominator we “agree” to
rationalize the denominator as follows:
a. If the denominator is a monomial, multiple the numerator and
denominator by the radical in the denominator.
b. If the denominator is a binomial, multiple the numerator and denominator by the conjugate
of the denominator. (The conjugate of a  b is a  b )
4. Reduce fractions outside the radical if possible.
1.
1
2.
5
5.
5a 2
2
1 2
3.
6.
9.
2x
5
5
64
7
6
6
3a
8.
9
7.
15 7
2 5
10.
2 3
1 5
4.
3
6
MAT1033
9.7 Complex Numbers
1  i
By definition:
So,
i 2  1
We express the square root of a negative number using a + bi form.
Example:
 16 written in this form is 4i
Example: 2 + 3  81 written in this form is 2 + 39i = 2+27i
 Add or subtract complex numbers. Write your answers in a + bi
form.
1. (3 + 2i) + (7 – 4i)
2. (5 – 3i) – (-2 + 4i)
 Multiply complex numbers. Write your answers in a + bi form.
Use i  1 when simplifying
2
3. 2i (3 – 5i)
4. (1 – i)(2 + 5i)
5. (3 + 2i) (3 – 2i)
6. (7 – 4i)2
 Divide complex numbers. Write your answers in a + bi form.
Use i  1 when simplifying
2
7.
2
1 i
8.
4
2  3i
 1  5i
9.
3  2i