Download Introduction to Radicals

Math 51
Worksheet
Introduction to Radicals
Radicals will be: evaluated, simplified, added, subtracted, multiplied and divided.
Square root or radical notation
index
⇒
Radicand
What number multiplied by itself gives the number under the radical?
• Perfect square numbers evaluate to rational numbers
25 = 5 Because 5 ⋅ 5 = 25
•
Other numbers require the use of a calculator and evaluate to irrational number
13 = 3.605551275…
•
Can you take the square root of a negative number? Why?
− 16 = ??? Does (−4)(−4) = −16 No. So, the answer is not a real number.
Square roots have an index of 2 other radicals are possible index 3, 4, 5 and so on. But the definition is the same.
3
125 = 5
5 ⋅ 5 ⋅ 5 = 125
Because,
or
4
81 = 3
Because,
3 ⋅ 3 ⋅ 3 ⋅ 3 = 81
and so on.
Simplifying radicals
x 6 = x 3 Why? Because x 3 ⋅ x 3 = x 6
What is the square root of a variable raised to a power?
6
2
6
x = x = x3
Another way to think about it is divide the power by the index.
If the power does not divide
evenly then the variable must be factored first so that the power can be divided by the index.
Example:
4
4
x
14
12
x x
14
4
=x
2
The power does not reduce to a whole number. So, factor the part that does:
⇒ x
12
4 4
x2
⇒ x3 4 x2
⇐ The remainder stays inside the radical.
Next Example: Simplify
108 x 5 y 8
Problem? 108 is not a perfect square but one of its factors maybe.
(36)(3) x 5 y 8
What about
(36)(3) x 4 x y 8
Everything that can be simplified comes out of the radical and everything else
x 5 5 divide 2 does not work, so that must be factored as well.
must stay inside.
4
(36)(3) x x y
8
4
2
6x y
⇒
8
2
3x ⇒ 6 x 2 y 4
3x
The same rules apply to all other radicals just the index changes.
Example:
3
7
135 x y
15
⇒
3
6
15
2
5 3
(27)(5) x x y
Answer:
3x y
⇒
5x
6
3
3x y
15
3 3
5x
Reduce all terms.
Multiply/Divide radicals: If the indexes are the same write as one radical.
Multiply
♦
4
2x 3 y ⋅ 4 8x 5 y 2
⇐ The indexes are the same “4”. So, write as one radical first.
♦
4
2x 3 y ⋅ 8x 5 y 2
⇐ Everything inside will multiply. Note: Remember your power rules.
♦
4
2 ⋅ 8x 3 x 5 y y 2
⇒
4
16 x 8 y 3
2 3 27 x is read as 2 times
Example:
3
⇒ Reduce ⇒
2x2 4 y 3
27 x When the radical is reduced whatever comes out of the radical is
multiplied by what is already out front.
♦
3
2 27 x ⇒ 2 ⋅ 3
3
x ⇒ 6
3
x Done!
Example: Multiply
5 x 4 x 3 ⋅ 6 x 2 18 x 4
⇐ Since this is multiplication we can reshuffle all terms.
5x ⋅ 6 x 2 4 x 3
⇒
30 x 3
30 x 3
⇒
30 x 3 ⋅ 6 x 3 2 x
3
162 x 7 y 10
18 x 4
36 ⋅ 2 x 6 x
72 x 7
⇒ Reduce the radical
⇒ 180 x 6
2 x Done!
Divide
♦
3
♦
3
⇐ The indexes are the same “3”. So, write as one big radical first.
6x 4 y
162 x 7 y 10
6x 4 y
⇐
Reduce
⇒
3
27 x 3 y 9
⇒ Simplify ⇒ 3xy 3
Adding/Subtracting radicals: The same as like terms. If all the terms match then combine the number out front.
Example:
8 11x + 5 11x − 7 11x
⇒
(8 + 5 − 7) 11x
⇒
6 11x
Some radicals may need to be reduced first before they are added.
Example:
♦
5 x 2 3 x + 4 12 x 5 − 2 x 27 x 3
⇐ Term do not match
♦
5 x 2 3x + 4 4 ⋅ 3x 4 x − 2 x 9 ⋅ 3x 2 x
⇐ Reduce each radical if possible
♦
5 x 2 3x + 4 ⋅ 2 x 2 3x − 2 ⋅ 3x ⋅ x 3x
⇐ Multiply terms out front
♦
5 x 2 3x + 8 x 2 3x − 6 x 2 3 x
⇐ Add like radicals ⇒
7 x 2 3 x Done!
Practice Problems
Perform the indicated operation and reduce all radicals.
1)
128x 3
2)
243m 5 n 2
3)
27a 2
4)
2⋅ 8
5) − 7 3 3 y 2 ⋅ 3 18 y
6)
7)
9)
4
3
25 p ⋅ 3 125 p 3
48 x 3 y 7
4x2
25 y 4
8)
8m 7 ⋅ 4 6m 3
10) 5 7 − 4 7 + 2 7
3 xy 3
12)
11) 3 x 7 + 28 x 2 − 63x 2
50 + 98 − 75 + 27
Answer Key
1)
8x 2x
2) 9m 2 n 3m
3)
3a 3
4)
4
5)
− 21 y
6)
5p
7)
2x
5y2
8)
4xy 2
9)
3
2
2m 2 4 3m 2
11) 2x 7
3
25 p
10) 3 7
12) 12 2 − 2 3